On Serre's Conjecture Over Imaginary Quadratic Fields
On Serre’s Conjecture Over Imaginary Quadratic Fields
Rebecca S. Torrey
A thesis submitted for the degree of Doctor of Philosophy at King’s College London, University of London
Department of Mathematics King’s College London Strand London WC2R 2LS United Kingdom
July 2009
For my mom
Declaration This thesis is a presentation of my own original research work. Wherever contributions of others are involved, every effort has been made to indicate this clearly with due reference to the literature.
Signature: ................................................................. Date: .............................. 3
Abstract In this thesis, I investigate a generalization to imaginary quadratic fields of the refined version of Serre’s conjecture. I ask whether a conjecture analogous to that of Buzzard, Diamond and Jarvis will hold over imaginary quadratic fields. I wrote code to compute cohomological mod ` modular forms over Q(i) of arbitrary weight. I adapt methods of Ash et al., which make use of Borel-Serre duality to express the relevant space of modular forms as a homology group, and then use modular symbols methods generalizing work of Cremona and others. Using an approach of Wiese, I prove algebraically that the modular symbols method will work in this more general setting. With data computed by my program, I provide evidence for Serre’s conjecture in this context.
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Acknowledgements First and foremost, I would like to take this opportunity to thank my supervisor, Fred Diamond. I am grateful to him for his support, for his unending patience and for sharing with me his wealth of knowledge in this beautiful subject. Other mathematicians whose assistance, advice and work have helped make this thesis possible include Avner Ash, Kevin Buzzard, John Cremona, David Dummit, Frazer Jarvis, Mehmet Haluk S¸eng¨ un and Gabor Wiese. I would also like to thank Manuel Breuning, David Burns and Payman Kassaei, and all the other excellent number theorists in London. It has been an honor to be a part of the London number theory community. My thesis is quite computational in nature, and I owe many thanks to the technical support team of the King’s College maths department, particularly to Dan Wade. I would also like to thank Paul Mitchell for providing computing resources for my code. I started my PhD at Brandeis University and I owe a great deal to many people there. In particular, I would like to thank Ruth Charney, Jennifer Johnson-Leung and Susan Parker. Most of all, I would like to thank my friends and family for their patience, love and support. I have made some great friends among my fellow PhD students, particularly Maria Alicia Lopez Osorio, Chunyan Mu, Aftab Pande and Mark Radosevich. Their humour and support are greatly appreciated. I would like to thank my Go teacher, Guo Juan, and all my Go playing friends. I give thanks to the many friends and family who came to visit me in London, providing little snippets of home, as well as all those who sent their love via post. There are also many old friends whose support from afar I greatly appreciate. I would particularly like to thank my parents Bruce 5
and Kathi and my siblings Mark, Kate, David and Emma and also the matriarch of our wonderful family, Grandma Eileen Noble. Finally, I would like to thank Rich Chalmers without whose patience, support and friendship this would not have been possible.
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Table of contents Declaration
3
Abstract
4
Acknowledgements
5
1 Introduction
10
2 Classical Serre’s Conjecture
13
2.1
Modular Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13
2.2
Galois Representations . . . . . . . . . . . . . . . . . . . . . . . . . .
15
2.2.1
Level of ρ . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
16
2.2.2
Character of ρ . . . . . . . . . . . . . . . . . . . . . . . . . . .
18
2.3 Serre’s Conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18
3 Generalized Serre’s Conjecture
21
3.1 Serre Weights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
21
3.2 Fundamental Characters . . . . . . . . . . . . . . . . . . . . . . . . .
24
3.3 The Weight Conjecture for Totally Real Fields . . . . . . . . . . . . .
26
3.4 Cohomological mod ` Forms Over K . . . . . . . . . . . . . . . . .
32
3.5 Serre’s Conjecture for Imaginary Quadratic Fields . . . . . . . . . . .
33
4 Examples of Galois Representations
36
4.1 Examples from Elliptic Curves . . . . . . . . . . . . . . . . . . . . . .
36
4.2 Examples from Polynomials . . . . . . . . . . . . . . . . . . . . . . .
39
7
4.2.1
Techniques & Background . . . . . . . . . . . . . . . . . . . .
39
4.2.2
Dihedral Group D4 . . . . . . . . . . . . . . . . . . . . . . . .
43
4.2.3
Alternating Group A4 . . . . . . . . . . . . . . . . . . . . . .
46
4.3 Examples from Class Field Theory . . . . . . . . . . . . . . . . . . .
52
4.3.1
Dihedral Group D3 . . . . . . . . . . . . . . . . . . . . . . . .
52
4.3.2
Dihedral Group D5 . . . . . . . . . . . . . . . . . . . . . . . .
56
5 Computing Modular Forms over Q(i)
59
5.1 Borel-Serre Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . .
59
5.2 An Algebraic Proposition . . . . . . . . . . . . . . . . . . . . . . . . .
62
5.3 Manin Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
71
5.4 Γ1 (n) and Characters . . . . . . . . . . . . . . . . . . . . . . . . . . .
77
5.5
79
Hecke Operators
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6 Computational Evidence
81
6.1
Algorithm for Computing Modular Forms
. . . . . . . . . . . . . . .
81
6.2
Torsion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
84
6.3
Modular Forms Corresponding to Elliptic Curves . . . . . . . . . . .
85
6.4
Modular Forms Corresponding to Representations from Polynomials .
90
6.4.1
Dihedral Group D4 . . . . . . . . . . . . . . . . . . . . . . . .
90
6.4.2
Alternating Group A4 . . . . . . . . . . . . . . . . . . . . . .
92
6.5 Modular Forms Corresponding to Representations from CFT . . . . .
95
6.5.1
Dihedral Group D3 . . . . . . . . . . . . . . . . . . . . . . . .
95
6.5.2
Dihedral Group D5 . . . . . . . . . . . . . . . . . . . . . . . .
99
6.6 Final Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 References
102
8
List of Tables 4.1
Weights for elliptic curves mod 7 . . . . . . . . . . . . . . . . . . . .
37
4.2
Elliptic curves over Q(i) mod 7 with several levels . . . . . . . . . .
39
4.3
D4 representation mod 5 with level n = 29 . . . . . . . . . . . . . .
45
4.4
A4 representation mod 3 with level n = 61 . . . . . . . . . . . . . .
48
4.5
A4 representation mod 3 with level n = 79 . . . . . . . . . . . . . .
51
4.6
D3 representations mod 5 and mod 7 with several levels . . . . . .
55
4.7
D5 representation mod 11 with level n = 19 + 20i . . . . . . . . . .
58
6.1 Elliptic curve, conductor n = 6 + 6i, considered mod 7 . . . . . . . .
86
6.2 Elliptic curve, conductor n = 9 + 7i, considered mod 7 . . . . . . . .
87
6.3 Elliptic curve, conductor n = 15, considered mod 7 . . . . . . . . . .
88
6.4 Elliptic curve, conductor n = 19 + 9i, considered mod 7 . . . . . . .
89
6.5
Galois group D4 , level n = 29, considered mod 5 . . . . . . . . . . .
91
6.6 Galois group A4 , level n = 61, considered mod 3 . . . . . . . . . . .
93
6.7
Galois group A4 , level n = 79, considered mod 3 . . . . . . . . . . .
94
6.8
Galois group D3 , level n = 8 + 17i, considered mod 5 and mod 7 .
96
6.9
Galois group D3 , level n = 13 + 28i, considered mod 5 and mod 7
97
6.10 Galois group D3 , level n = 8 + 35i, considered mod 5 and mod 7 .
98
6.11 Galois group D5 , level n = 19 + 20i, considered mod 11 . . . . . . . 100
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Introduction The Langlands Program, started in the late 1960s by Robert Langlands, is a system of powerful conjectures connecting number theory to the representation theory of certain groups. It incorporates an earlier construction of Shimura which associates Galois representations to modular forms. Serre’s conjecture, first formulated by Jean-Pierre Serre in a 1987 article [Ser87], provides a converse in characteristic ` to Shimura’s construction. Serre’s conjecture states that any odd, irreducible representation ρ : Gal(K/Q) → GL2 (Fq ) where K is a Galois number field and Fq is a finite field of characteristic `, is modular, i.e., arises from a modular form. A refinement to Serre’s conjecture gives the minimal weight and level of this modular form and, by work of Ribet and others, we know that Serre’s conjecture holds if and only if the refined version holds. Recently, Khare and Wintenberger have completed the proof of Serre’s conjecture using ideas and results of Dieulefait, Kisin, Taylor and Wiles. It is natural to ask whether an analogous conjecture holds for representations of Gal(K/F ) where F is an arbitrary number field. Buzzard, Diamond and Jarvis [BDJ] recently formulated a version of the refined Serre’s conjecture in the case in which F is totally real and ` is unramified, where predicting the weights is much more complicated than for F = Q. In this more general setting, it no longer makes sense to simply specify a minimal weight and level for the corresponding modular form. A more general notion of weight is needed. Also, the refined part of the conjecture takes the form of a recipe for all the weight combinations for modular forms giving rise to a particular representation. This recipe depends only on the local behavior of the representation ρ at primes above `. There has been some progress, due to Gee, towards proving the equivalence of Serre’s conjecture and the refined Serre’s conjecture for totally real fields F , but crucial steps in the method of Khare 10
and Wintenberger break down in this situation. If F is not totally real, then the situation is even less well understood. In this case, we do not even have a complete understanding of how to associate Galois representations to modular forms. This situation is the focus of my research. Some computational evidence is available for generalizations of Serre’s conjecture to number fields. Demb´el´e [DDR] has done computations of arbitrary weight mod ` modular forms over totally real fields F , providing evidence for the conjecture of Buzzard, Diamond and Jarvis. For imaginary quadratic fields, Figueiredo [Fig99] provided some computational evidence for Serre’s conjecture, but he worked only with weight two modular forms. More recently, S¸eng¨ un [S¸08] proved non-existence of certain representations and has done some computations of arbitrary weight modular forms over imaginary quadratic fields. In this thesis, I ask whether a conjecture analogous to that of Buzzard, Diamond and Jarvis will hold over imaginary quadratic fields. I wrote code to compute cohomological mod ` modular forms over Q(i) of arbitrary weight. My approach differs from that of S¸eng¨ un, who computes cohomology. Instead, I compute homology using modular symbols methods generalizing work of Cremona [Cre84] and others. This homological method seems to be more promising for adapting methods developed by Cremona and others for computing forms over imaginary quadratic fields of class number greater than one. In Chapter 2, I review the classical case, i.e. Serre’s conjecture over Q. I give basic definitions and some basic facts about modular forms and Galois representations in this setting. I state Serre’s original conjecture and discuss work that has been done on the conjecture itself and towards generalizations of the conjecture. In Chapter 3, I define Serre weights and review the conjecture of Buzzard, Diamond and Jarvis (BDJ) over totally real fields. I then define mod ` modular forms cohomologically for imaginary quadratic fields and ask whether the BDJ conjecture will hold in the imaginary quadratic case. I compute examples of Galois representations in Chapter 4. Some of these examples arise from polynomials, some are from elliptic curves and others are constructed using class field theory. For each I find the level and character and then compute the 11
predicted weights using the BDJ conjecture. I compute the predicted weights for two of the three examples given in Figueiredo’s thesis. In Chapter 5, I prove that the modular symbols method can be used to compute the space of modular forms in which I am interested. To account for the higher weights, one needs to compute cohomology with non-trivial coefficients. In this case, Cremona’s geometric approach becomes intractable. I adapt methods of Ash et al., which make use of Borel-Serre duality to express the relevant space of modular forms as a homology group. From Borel-Serre [BS73], we have an isomorphism ∼
H 2 (Γ, V ) −→ H0 (Γ, St ⊗ V ), where St denotes the Steinberg module, and V denotes a “Serre weight”, i.e., an ¯ ` -representation of irreducible F G = GL2 (OK /`OK ). Following Ash in [Ash94], I describe the Steinberg module in terms of universal minimal modular symbols. This allows one to sidestep the geometric argument, and prove algebraically that the modular symbols method can be used to compute H0 (Γ, St⊗V ). In order to obtain a description of the space which can actually be used for computations, one must be able to go from modular symbols to Manin symbols (M-symbols). Without recourse to Cremona’s geometric method, I needed a different way to justify this conversion. Following an approach of Wiese [Wie05], I prove algebraically that one can use M-symbols to compute this homology space for K = Q(i). In Chapter 6, I present computational evidence in support of the conjecture. I summarize the algorithm used in my program to compute the cohomological mod ` modular forms over Q(i). Finally, I give tables of systems of eigenvalues matching the traces of ρ(Frobp ) of the Galois representations ρ computed in Chapter 4.
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Classical Serre’s Conjecture Serre’s conjecture relates Galois representations to modular forms. In this chapter, we set the stage by reviewing the classical case. We will give basic definitions of modular forms and Galois representations and then state Serre’s original and refined modularity conjectures. Finally we give a brief overview of the current status of Serre’s conjecture (which is now a theorem) and indicate various work undertaken towards understanding the more general situation.
2.1
Modular Forms
We start by giving the classical definition of modular forms (i.e., over Q). Modular forms are functions on the upper half of the complex plane which satisfy certain symmetry and “boundedness” conditions. To describe these conditions in more detail, we first need some notation. Let h = {z ∈ C | Im(z) > 0} be the complex upper half plane. The modular group (Ã SL2 (Z) =
a b c d
!
) | a, b, c, d ∈ Z and ad − bc = 1 Ã
acts on h by fractional linear transformations: for g = z ∈ h, we have g(z) =
az + b . cz + d
13
a b c d
! ∈ SL2 (Z), and
Any subgroup Γ of SL2 (Z) which contains the subgroup (Ã Γ(N ) =
a b
!
à ∈ SL2 (Z) |
c d
a b
!
à ≡
c d
1 0
!
) mod N
0 1
for some positive integer N is called a congruence subgroup. The level of a congruence subgroup Γ is the smallest N such that Γ(N ) ⊆ Γ. We define two important congruence subgroups in particular: (Ã Γ0 (N ) =
a b
!
à ∈ SL2 (Z) |
c d
a b
!
à ≡
c d
∗ ∗
!
) mod N
0 ∗
and (Ã Γ1 (N ) =
a b c d
!
à ∈ SL2 (Z) |
a b
!
c d
à ≡
∗ ∗ 0 1
!
) mod N
.
Definition 2.1.1. A modular form of weight k and level N is a function on h which is holomorphic everywhere (including at the cusps) and which satisfies f (z) = (cz + d)−k f (g(z)), Ã for all g =
a b c d
! in Γ, where Γ is a congruence subgroup of level N .
The modular forms of weight k for a congruence subgroup Γ form a finite dimensional complex vector space which we denote by Mk (Γ). Ã ! 1 1 For Γ = Γ0 (N ) or Γ = Γ1 (N ), we have ∈ Γ, so that a modular form f 0 1 for such a congruence subgroup satisfies f (z) = f (z + 1). This, together with the holomorphicity condition, implies that f has a Fourier ex-
14
pansion of the form f (z) =
∞ X
an q n ,
q = e2πiz .
n=0
à (For any congruence subgroup Γ, we at least have
1 N
!
∈ Γ, giving a similar 0 1 expansion but with q = e2πiz/N .) If a0 = 0 in the Fourier expansion of (cz+d)−k f (g(z)) for all g ∈ SL2 (Z), we call f a cusp form. We denote the space of cusp forms by Sk (Γ). For Γ = Γ1 (N ), one can define certain operators Tn for n ≥ 1, called Hecke operators, and hdi for d ∈ (Z/N Z)∗ , called diamond bracket operators, which act on the space Mk (Γ1 (N )) and on the space Sk (Γ1 (N )). Together these operators form a commutative algebra called the Hecke algebra H = Z[Tn , hdi]. A non-zero cusp form f ∈ Sk (Γ1 (N )) is called an eigenform if it is a simultaneous eigenvector for all operators in the Hecke algebra. If f ∈ Sk (Γ1 (N )) is an eigenform, we define the character of f to be the Dirichlet character ε : (Z/N Z)∗ → C∗ such that f |hdi = ε(d)f . We denote the space of forms with a given character ε by Sk (Γ1 (N ), ε). We define a system of eigenvalues in Sk (Γ1 (N ), ε) to be a set of eigenvalues (an ) of the Hecke operators Tn for n ≥ 1 acting on an eigenform f ∈ Sk (Γ1 (N ), ε). One can show that the following relationships hold for Hecke operators: 1. Tmn = Tm Tn
if gcd(m, n) = 1, and
2. Tpn = Tpn−1 Tp − pk−1 ε(p)Tpn−2
for p prime.
Thus to determine a given system of eigenvalues one need only compute the eigenvalues ap for p prime.
2.2
Galois Representations
The Galois representations of Serre’s Conjecture (and, consequently, those that will be of concern to us) are the mod ` Galois representations. Recall the definition:
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Definition 2.2.1. A mod ` Galois representation is a continuous homomorphism ¯ ` ). ρ : GQ −→ GLn (F In this thesis, the representations that we will be interested in will be continuous homomorphisms ¯ `) ρ : GK −→ GL2 (F ¯ where K is an imaginary quadratic field and GK = Gal(Q/K). Since we are interested in the refined version of Serre’s conjecture, we will also need the definitions of level, weight and character. In the following two subsections we define the level and character associated to ρ. The definition of the weight kρ is more complicated. Instead of defining it specifically for the classical case, we will discuss the BDJ generalization of the weight recipe in Chapter 3.
2.2.1 Level of ρ To define the level of ρ, we follow Serre’s exposition in his seminal 1987 paper on the subject ([Ser87]). The level is defined to be the Artin conductor of ρ, defined as it is in characteristic 0, except that in this case we take the prime to ` part of the conductor. We will now give the precise definition. First write the Galois representation as ρ : GQ −→ GL(V ), ¯ ` . Let p 6= ` be a rational prime. The where V is a 2-dimensional vector space over F representation ρ will factor through some finite extension L over Q, so we may write ρ : Gal(L/Q) −→ GL(V ). Let G be the Galois group Gal(L/Q). Let D0 ⊃ D1 ⊃ · · · ⊃ Di ⊃ · · · 16
be the higher ramification groups of G corresponding to the prime p. For each i, denote by Vi the subspace of V fixed by Di . Define the integer n(p, ρ) by n(p, ρ) =
∞ X i=0
1 dim(V /Vi ). (D0 : Di )
We have the following properties regarding n(p, ρ). 1. n(p, ρ) = 0 if and only if D0 is trivial, which is if and only if ρ is unramified at p; 2. n(p, ρ) ≥ 1 if ρ is ramified at p; 3. n(p, ρ) = dim(V /V0 ) if and only if D1 is trivial, which is if and only if ρ is tamely ramified at p. We define the level N (ρ) by N (ρ) =
Y
pn(p,ρ) .
p6=`
This thesis is concerned with representations over an imaginary quadratic field K, i.e. ρK : GK → GL(V ). Now ρK will factor through some Galois extension L over K, so the Galois group in question will be G = Gal(L/K). From there on, we can define the level the same way, except that we will take primes p ∈ OK , so that the level n(ρK ) =
Y
pn(p,ρ)
p-`·OK
will be an ideal of OK . The level will still, by definition, be prime to `.
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2.2.2 Character of ρ In this section, we will define a generalization to imaginary quadratic fields K of the character ερ that Serre associated to ρ in [Ser87]. This character ερ will be obtained from the character ¯ ¯ `) det ρ : GK = Gal(Q/K) → GL2 (F by removing the ` part. We will now make this more precise. Since the conductor of det ρ must divide `n(ρ) (see, e.g., [Fig95, p 7]), we can identify det ρ with a homomorphism ¯ ∗, det ρ : (OK /`n(ρ))∗ → F ` or, equivalently, to a pair of homomorphisms ¯∗ ϕρ : (OK /`OK )∗ → F ` and ¯ ∗. ερ : (OK /n(ρ))∗ → F ` We define the character of ρ to be the character ερ .
2.3
Serre’s Conjecture
In a 1987 paper [Ser87], Serre gave a conjecture prescribing a relationship between modular forms and mod ` Galois representations. Conjecture 2.3.1. (Serre, 1987) Suppose ¯ ` ), ρ : GQ → GL2 (F is a
mod ` Galois representation which is continuous, odd and irreducible. Then P there is a modular form f = an q n such that tr(ρ(Frobp )) = ap and det(ρ(Frobp )) = ε(p)pk−1 for all p - N `. Here N is the level, k the weight and ε the character of f . 18
Whenever such an f does exist for a given ρ, we say that ρ is modular. Furthermore, Serre gave a refinement of his conjecture, in which he prescribed the minimum weight and level of such an eigenform f , assuming k ≥ 2 and ` - N , using the definitions given in the preceding section (though we have omitted the definition of the weight kρ ). Conjecture 2.3.2. (Serre, refined) Suppose ¯ `) ρ : GQ → GL2 (F is a
mod ` Galois representation which is continuous, odd and irreducible. Then ρ
is modular and one can take the corresponding modular form f to be in Skρ (Γ, ερ ) with Γ of level N (ρ). Furthermore, the level N (ρ) and the weight kρ are minimal among levels prime to ` and weights k ≥ 2. Serre’s conjecture is now a theorem. First, work by Ribet, Gross, Coleman-Voloch and others showed that the original conjecture and the refined conjecture are equivalent (see, e.g., [Dia97]). Recently Khare and Wintenberger ([KWa], [KWb]) proved the conjecture itself using work of Dieulefait, Taylor, Wiles and Kisin. This thesis is concerned with a generalization of Serre’s conjecture. There are two natural avenues to explore. One is to consider representations of higher dimension, i.e. representations ¯ ` ), ρ : GQ → GLn (F for n > 2. Progress has been made in developing conjectures and providing evidence for these conjectures by Ash, Doud, Pollack and Sinnott (see [ADP02] and [AS00]) and also by Herzig ([Her]). In Chapter 4 we will look at some examples of 2-dimensional representations considered in the papers by Ash et al. They use these examples to construct reducible 3-dimensional representations, but we will consider irreducible 2-dimensional representations they compute in the process. A second natural generalization of Serre’s conjecture is to consider more general
19
number fields, i.e. representations ¯ ) → GL2 (F ¯ ` ), ρ : GF = Gal(Q/F for an extension F of Q. This type of generalization is the focus of this thesis. In a forthcoming paper, Buzzard, Diamond and Jarvis (BDJ) [BDJ] consider the case of totally real fields F . They formulate a version of Serre’s refined conjecture in this context, which we will review in Section 3.3. Totally real fields are a natural starting place as there is already a complete map in the other direction, i.e. one can associate a Galois representation to a Hilbert modular cusp form. In this situation, it no longer makes sense to talk about minimal weights. Instead ¯ ` representations Buzzard, Diamond and Jarvis define Serre weights to be irreducible F σ of GL2 (O/`O) and explain what it means for a representation to be modular of weight σ. Their conjecture then, instead of giving a minimal weight, describes all the possible Serre weights and level structures of modular forms giving rise to a given representation ρ. They show that their conjectural weight recipe is correct for K = Q. They also cite both theoretical and computational evidence supporting the conjecture. This includes calculations by Demb´el´e, Diamond and Roberts [DDR] of Hilbert modular forms with these weights and level structures and their corresponding Galois representations. The BDJ conjecture covers mod ` representations for unramified primes `. Schein [Sch] has subsequently extended this conjecture to ramified primes ` > 2. In [Gee07] and [Gee], Gee made some progress towards proving the regular and refined versions of Serre’s conjecture are equivalent in the totally real case. In a 1998 paper [Fig99], Figueiredo investigated the possibility of generalizing Serre’s Conjecture to imaginary quadratic fields. He computed
mod ` cusp forms
and corresponding Galois representations, providing evidence that an analogous correspondence would also hold in this situation. His computations are limited to weight 2 and he did not formulate a refined version of Serre’s Conjecture. In this thesis we explore the refined version of Serre’s conjecture over imaginary quadratic fields. 20
Generalized Serre’s Conjecture In this chapter, we discuss the generalization of Serre’s conjecture to number fields. We define Serre weights, a generalization of the classical notion of weights. We review some basics about fundamental characters, which will be used in subsequent sections. We discuss the BDJ conjecture for totally real fields. Finally we define cohomological mod ` forms for imaginary quadratic fields and ask whether the BDJ conjecture will hold in this case as well.
3.1
Serre Weights
Let K be a number field and denote by O its ring of integers. Fix a prime ` which is unramified in K. ¯ ` -representation V of Definition 3.1.1. Define a Serre weight to be an irreducible F G = GL2 (O/`O). In this section we describe what these Serre weights look like and describe the form in which we work with them computationally. Denote by SK the set of embeddings τ : K ,→ C. Set G = GL2 (O/`O) ∼ =
Y
GL2 (O/p).
p|`
For each prime p of K such that p|`, set kp = O/p and fp = [kp : F` ]. We may identify Q ¯ ` . We will see that the SK with p|` Sp , where Sp is the set of embeddings kp ,→ F ¯ ` -representations of G are of the form irreducible F V =
O
Vp
¯ `) (the tensor product is taken over F
p|`
21
where Vp =
O¡
¢ ¯ `. detaτ ⊗kp Symbτ −1 kp2 ⊗τ F
τ ∈Sp
¯ ` -representation V is a tensor product over primes p dividing `; each Note that the F factor Vp will act on the corresponding factor GL2 (O/p) of GL2 (O/`O). For each of these factors Vp , we take the tensor product over all τ ∈ Sp , so we will take a closer look at these embeddings τ . Fix some τ0 ∈ Sp : ¯ `. τ0 : kp = O/p ,→ F Then for each i such that 1 ≤ i < f where f = [kp : F` ], set τi = τ0 ◦ Frobi` . We then have Sp = {τi : 0 ≤ i < f }. For computational purposes, we will think of Symbτ −1 (kp2 ) as the space of homogeneous polynomials of degree bτ − 1 in two variables with coefficients in kp . We define the left action of GL2 (OK ) on this space as follows: for g ∈ GL2 (OK ), we reduce g modulo p and then the action is given by g¯ · P (X, Y ) = P (dX − bY, −cX + aY ), Ã for g¯ =
a b
c d the inverse, i.e.,
! ∈ GL2 (OK /p). We will also need a right action, for which we take
P (X, Y ) · g¯ = (¯ g )−1 · P (X, Y ). Since the Serre weights are irreducible representations, we may assume 1 ≤ bτ ≤ `. If bτ > `, the ideal I generated by X ` and Y ` is stable under the action of G since g(X ` ) ∈ I and g(Y ` ) ∈ I so g(I) ⊆ I. In this case, we get a G-stable subspace of ¯ 2 ), so this is a reducible representation of G. Symbτ −1 (F `
The determinant map is simply multiplication by the determinant (or some power of the determinant) of g for g ∈ G. This is sometimes denoted detaτ kp2 . We can view 22
the representation
O
¯ `) (detaτ ⊗τ F
τ ∈Sp
¯ ` via the embedding (where the inner tensor product is taken over the image of kp in F by that particular τ ) as the tensor product of characters ¯× (τ ◦ det)aτ : GL2 (kp ) −→ F ` for integers aτ . That is, for each embedding τ , we have the diagram ¯× GL2 (kp ) −→ F ` ∪ .
det ↓ τ
−→ F× `f
kp× So now we can write
O
¯ `) = (detaτ ⊗τ F
τ
O
(τ ◦ det)aτ .
τ
Furthermore, writing the various τ ∈ Sp as τi (as defined above), we can express this as
O
(τ ◦ det)
aτ
=
τ
f −1 O
(τi ◦ det)ai ,
i=0
¯ ` . Then where ai = aτi and both tensor products are taken over F f −1 O
f −1 ai
(τi ◦ det) =
i=0
Y
i
(τ0 ◦ det)ai ` = (τ0 ◦ det)
Pf −1 i=0
a i `i
.
i=0
Summarizing, we get that O
¯ ` ) = (τ0 ◦ det) (detaτ ⊗τ F
Pf −1 i=0
a i `i
.
τ
Thus the representation
N
aτ ¯ ` ) is determined by the exponents Pf −1 ai `i (det ⊗ F τ i=0 τ
mod (`f − 1), so we can now see that to get all of these representations, it suffices to allow the ai to run from 0 to ` − 1. Taking all the fp -tuples of ai where 0 ≤ ai ≤ ` − 1, 23
we get all the distinct representations of this type, with one redundancy: if all the ai are 0 we get the same representation as when all the ai are ` − 1. Thus we make the additional assumption that in each fp -tuple of ai , one of the ai is strictly less than ` − 1.
3.2
Fundamental Characters
In this section we review the concept of fundamental characters, as we will need this in the subsequent section where we define the conjectural weight recipe. Let K be a local field of characterstic 0, so K is a finite extension of Qp for some rational prime p. Let L be a finite Galois extension of K and let l and k be the residue fields of L and K respectively. There is a natural surjection of Galois groups Gal(L/K) → Gal(l/k). We define the inertia group of L over K, denoted I(L/K), to be the kernel of this map. For an extension L0 over L we have a surjection I(L0 /K) → I(L/K) and so we can define the inertia group IK to be the inverse limit of the I(L/K) as L ranges over ¯ all finite Galois extensions of K in K. From field theory, we know that for each n there is an extension of k of degree n
n, which we will denote by kn . This extension is the splitting field of X q − X and it is unique up to k-isomorphism. The Galois group of this extension, Gal(kn /k), is cyclic of order n, with canonical generator the Frobenius element x 7→ xq . Milne’s Proposition 7.50 [Mil08, p.121] says the following: Proposition 3.2.1. Let L be an algebraic extension of K. There is a one-to-one inclusion reversing correspondence of sets {K 0 ⊂ L, finite and unramified over K} ↔ {k 0 ⊂ l, finite over k},
24
where k 0 is the residue field of K 0 . Furthermore, there is a canonical isomorphism Gal(K 0 /K) → Gal(k 0 /k). From this proposition and the preceding discussion, we see that for each n there is n
a degree n unramified extension Kn of K. It is again the splitting field of X q −X and it is unique up to K-isomorphism. Since the Galois groups must be isomorphic, we see that Gal(Kn /K) is cyclic of order n. As before the Galois group has as canonical generator the Frobenius element σ which is the element of the Galois group satisfying the property σβ ≡ β q
mod p for all β ∈ OKn .
In subsequent sections, we will need something from local class field theory called ¯ ,→ K ¯ p . This gives us a corresponding a fundamental character. Fix an embedding K inclusion of Galois groups ¯ p /Kp ) ,→ Gal(K/K), ¯ Gal(K
σ 7→ σ|K¯ .
Via this inclusion, we may regard the inertia subgroup IKp as a subgroup of GK = ¯ Gal(K/K). Letting f denote the degree of the extension Kp over Q` , we have that 1
the residue field kp of Kp is isomorphic to F`f . Let Kp0 = Kp ((−`) `f −1 ). The field Kp contains all the (`f − 1)st roots of unity, so the extension Kp0 is Galois over Kp . We also know, from Kummer theory, that Gal(Kp0 /Kp ) = kp× = F× . `f Since Kp is unramified over Q` , the inertia group IKp injects into the Galois group ¯ ` /Kp ) which, in turn, injects into Gal(Q ¯ ` /Q` ). We thus get a map Gal(Q ¯ ` /Kp ) → Gal(K 0 /Kp ) → F×f , IKp ,→ Gal(Q p ` ¯ ` /Kp ) → Gal(K 0 /Kp ) is simply restriction to K 0 . The map where the map Gal(Q p p given above is called the fundamental character. IKp → F× `f 25
3.3
The Weight Conjecture for Totally Real Fields
In this section, we will describe, at least in part, the conjectural weight recipe given by Buzzard, Diamond and Jarvis in [BDJ] for the generalization to totally real fields of the refined version of Serre’s conjecture. Let K be a totally real field and fix a prime ` which is unramified in K. Let ¯ ` ) be a continuous, irreducible and totally odd representation. ρ : GK → GL2 (F As described in Section 3.1, each Serre weight V is of the form V = ⊗F¯` Vp where each Vp is an irreducible representation of GL2 (kp ) and the p run over all the primes of K which divide `. We will describe a set Wp (ρ) of GL2 (kp )-representations; the conjectural weight set W (ρ) will then consist of Serre weights of the form V = ⊗F¯` Vp where each Vp ∈ Wp (ρ). First, we will need some more notation. For the moment, we will be working with a fixed prime p dividing `, so we will suppress the p from the notation, i.e., instead of kp , fp and Sp , we will simply write k, f and S, respectively. We fix an ¯ ,→ K ¯ p and identify GKp and IKp with subgroups of GK . Denote by embedding K ¯ p , and extend this prime notation K 0 the unramified quadratic extension of Kp in K p
to its associated entities, i.e., we have k 0 , f 0 and S 0 . (We only need to consider the quadratic extension because we are only looking at 2-dimensional representations.) Write D = GKp and D0 = GKp0 . Define a projection map π : S 0 → S by τ 0 7→ τ 0 |k . For an embedding σ ∈ S or σ ∈ S 0 , let ωσ denote the fundamental character of IL defined by composing σ with the homomorphism from local class field theory IL → (O/`O)× (here L is either Kp or Kp0 depending on whether σ is in S or S 0 ). The sets Wp (ρ) will only depend on the local behaviour of ρ at ` and we will split the definition into two cases: when ρ|D is reducible and when it is irreducible. First we consider the case where ρ|D is irreducible. For a subset J of S 0 such that ∼
π : J → S, we define a matrix à Q M~b,J =
b
0
π(τ ) τ 0 ∈J ωτ 0
Q
!
0 b
0
π(τ ) τ 0 6∈J ωτ 0
0
26
.
Then we define Wp (ρ) by ( Wp (ρ) =
V~a,~b : ρ|I ∼
Y
) ∼
ωτaτ M~b,J for some J ⊂ S 0 such that π : J → S
.
τ ∈S
In this case, when ρ|D is irreducible, we can write ρ|D as the induction of a character, ¯ × . We use this to define a set W 0 (ξ) i.e., ρ|D ∼ IndD0 ξ for some character ξ : D0 → F D
`
which is very close to Wp (ρ): ( W 0 (ξ) =
∼
(V~a,~b , J) : J ⊂ S 0 , π : J → S, ξ|I =
Y
ωτaτ
τ ∈S
Y
) b 0) ωτπ(τ 0
.
τ 0 ∈J
We have Wp (ρ) = {V |(V, J) ∈ W 0 (ξ) for some J} and so we have a projection map ∼
W 0 (ξ) → Wp (ρ). We have another projection W 0 (ξ) → {J ⊂ S 0 |π : J → S}. Both of these projections are usually bijections, so that |Wp (ρ)| ∼ 2f . (Note that, in the above, if we replace the character ξ by its conjugate under D/D0 , this simply replaces J by its complement in S 0 .) Choose some τ00 ∈ S 0 and set τi0 = τ00 ◦ Frobi` and τi = π(τi0 ). Then we have S = {τi | i ∈ Z/f Z} and S 0 = {τi0 | i ∈ Z/2f Z}. Denote the fundamental characters for our fixed τ0 and τ00 by ω = ωτ0 and ω 0 = ωτ00 . Then we can write the other i
i
fundamental characters in terms of these two, namely, ωτi = ω ` and ωτi0 = (ω 0 )` . We also have ω = (ω 0 )`
f +1
. Note that ξ|I = (ω 0 )n for some n mod `2f − 1 and ρ|D
irreducible implies that (`f + 1) does not divide n. The following proposition, which is a combination of Propositions 3.1 and 3.2 in [BDJ], analyzes the combinatorics of the situation to make more precise the notion that |Wp (ρ)| is close to 2f . Proposition 3.3.1. Suppose ` is an odd prime and ξ, W 0 (ξ) and n are defined as above. Define A to be the set of congruence classes
mod `f + 1 of the form
−1 + (` + 1) P
if f is even
(` + 1) P
if f is odd,
i i i∈B ∗ (−1) `
i i i∈B ∗ (−1) `
27
where, in both cases, B ∗ runs over all non-empty proper subsets of {0, 1, . . . , f − 1}. The conjugacy classes of A are distinct and non-zero. Furthermore, we have
|W 0 (ξ)| =
2f
n 6∈ A
2f − 1 n ∈ A.
On the other hand, suppose ` = 2. Then 2f − 1 |W 0 (ξ)| = 2f 2f − 3
if f is even if f is odd and 3 - n if f is odd and 3 | n.
We review part of the proof as it will be useful in computing the predicted weights. Define n0a,~b,B = a(`f + 1) +
X
bi `i +
i∈B
X
bi `f +i
mod `2f − 1,
i6∈B
where a ∈ Z/(`f − 1)Z, ~b = (b0 , . . . , bf −1 ) with 1 ≤ bi ≤ ` and B ⊂ {0, . . . , f − 1}. We then have a bijection of sets W 0 (ξ) ↔ {(a, ~b, B) | n ≡ n0a,~b,B
mod `2f − 1}.
For each subset B of {0, . . . , f − 1}, there is a bijection between such triples (a, ~b, B) and solutions of the congruences n≡
X i∈B
bi `i −
X
bi `i
mod `f + 1,
(3.1)
i6∈B
with all the bi in {1, . . . , `}. But the values of the expression on the right hand side, P P i i f 0 f 0 i∈B bi ` − i6∈B bi ` , consist of the ` consecutive integers from nB − ` to nB − 1,
28
where n0B =
X
`i+1 −
i∈B
X
`i + 1.
i6∈B
So long as n 6≡ n0B mod `f + 1, we get a unique solution to the congruence (3.1) and so we have a bijection of sets mod `f + 1}.
W 0 (ξ) ↔ {B | n 6≡ n0B ∼
Also, the projection W 0 (ξ) → {J ⊂ S 0 | π : J → S} is injective. Buzzard, Diamond and Jarvis also give the following proposition, which gives criteria for determining when multiple B occur with the same (a, ~b). Proposition 3.3.2. (Proposition 3.3 in [BDJ]) The projection map from W 0 (ξ) to Wp (ρ) fails to be injective if and only if `r n ≡ m mod `f + 1 for some integers r, m with |m| ≤ `(`f −2 − 1)/(` − 1). Now consider the case where ρ|GKp is reducible. We write à ρ|GKp ∼
χ1
∗
0
χ2
! .
We define a set W 0 (χ1 , χ2 ), depending on these two characters, as follows: ( W 0 (χ1 , χ2 ) =
(V~a,~b , J) : J ⊂ S, χ1 |IKp =
Y τ ∈S
ωτaτ
Y
ωτbτ , χ2 |IKp =
τ ∈J
Y
ωτaτ
τ ∈S
Y
) ωτbτ
.
τ 6∈J
The weight set Wp (ρ) will be defined as a subset of the projection π1 (W 0 (χ1 , χ2 )) onto ! Ã χ1 ∗ , let cρ be the corresponding class the first component. Writing ρ|D ∼ 0 χ2 0 in H 1 (Kp , χ1 χ−1 2 ). To a given pair α = (V~a,~b , J) ∈ W (χ1 , χ2 ), Buzzard, Diamond ¯ ` (χ1 χ−1 )) and then define the and Jarvis associate a certain subspace Lα ⊂ H 1 (Kp , F 2
weight set by: Wp (ρ) = {V~a,~b : cρ ∈ Lα for some α = (V~a,~b , J) ∈ W 0 (χ1 , χ2 )}. 29
See [BDJ] for the definition of the subspace Lα . We analyze the set W 0 (χ1 , χ2 ) combinatorally, similar to the analysis in the irreducible case above. We will see that the projection π2 : W 0 (χ1 , χ2 ) → {J ⊂ S} is usually a bijection (and otherwise not far off) so that |W 0 (χ1 , χ2 )| ∼ 2f . Note also that interchanging χ1 and χ2 in W 0 (χ1 , χ2 ) replaces J by its complement. Find n1 and n2 in Z/(`f − 1)Z such that χ1 = ω n1 , and χ2 = ω n2 , and set n = n1 − n2 . The following three propositions are Propositions 3.4, 3.5 and 3.6 in [BDJ]. Proposition 3.3.3. Suppose that ` > 3. Define A to be the set of congruence classes mod `f − 1 of the form −1 + (` + 1) P ∗ (−1)i `i i∈B P (` + 1) (−1)i `i
if f is odd if f is even,
i∈B ∗
where, in the first case B ∗ runs over all subsets of {0, 1, . . . , f − 1}, and in the second case B ∗ runs over all non-empty proper subsets of {0, 1, . . . , f − 1}. The elements of A are distinct and non-zero. Furthermore, we have 2f + 2 |W 0 (χ1 , χ2 )| = 2f + 1 f 2
if n = 0 and f is even, if n ∈ A, otherwise.
Proposition 3.3.4. Suppose that ` = 3. Define A to be the set of congruence classes mod (3f − 1)/2 of the form −1 + 4 P ∗ (−1)i 3i i∈B 4 P (−1)i 3i i∈B ∗
30
if f is odd if f is even,
where, in the first case B ∗ runs over all subsets of {0, 1, . . . , f − 1} other than {0, 2, . . . , f − 1} and {1, 3, . . . , f − 2}, and in the second case B ∗ runs over all nonempty proper subsets of {0, 1, . . . , f −1} other than {0, 2, . . . , f −2} and {1, 3, . . . , f − 1}. The elements of A are distinct and non-zero. Furthermore, we have 2f + 2 |W 0 (χ1 , χ2 )| = 2f + 1 2f
if n = 0 and f is even, or n = (3f − 1)/2 if n ∈ A, otherwise.
Proposition 3.3.5. Suppose ` = 2. Then 2f 2f |W 0 (χ1 , χ2 )| = 2f 2f 2f
+4
if n = 0 and f is even,
+3
if n 6= 0, 3 | n and f is even,
+2
if n = 0 and f is odd,
+1
if n 6= 0 and f is odd, if 3 - n and f is even.
We recall part of the proof of the above propositions, as we will use the techniques to compute the weights in the reducible case. Define n0a,~b,B = a +
X
bi `i
mod `f − 1
i∈B
for a ∈ Z/(`f − 1)Z, ~b = (b0 , . . . , bf −1 ), with 1 ≤ bi ≤ `, and B ⊂ {0, . . . , f − 1}. ¯ denote the complement of B in {0, . . . , f − 1}. We look for triples (a, ~b, B) as Let B above such that n1 ≡ n0a,~b,B
mod (`f − 1) and
n2 ≡ n0a,~b,B¯
mod (`f − 1). 31
Such triples are in bijection with W 0 (χ1 , χ2 ). We can instead look for solutions of n≡
X
bi `i −
i∈B
X
bi `i
mod `f − 1.
i6∈B
For a particular subset B ⊂ {0, . . . f − 1}, there is a unique triple (a, ~b, B) for each solution of the above. We can use nB =
X
`i+1 −
i∈B
X
`i
i6∈B
to determine the number of solutions. In particular the values of
P
i i∈B bi ` −
P
i i6∈B bi `
are precisely the `f consecutive integers from nB + 1 − `f to nB . Thus if n 6≡ nB mod `f − 1, then we have a unique solution but if n ≡ nB mod `f − 1, then we have two solutions. Buzzard, Diamond and Jarvis also give the following proposition, which gives a criterion for determining when multiple B occur with the same (a, ~b). Proposition 3.3.6. (Proposition 3.7 in [BDJ]) The projection map from W 0 (χ1 , χ2 ) onto its first component fails to be injective if and only if `r n ≡ m mod `f − 1 for some integers r, m with |m| ≤ max{0, `(`f −2 − 1)/(` − 1)}. Note in particular that if n = 0 (e.g., in the case where ρ restricted to the decomposition group is trivial), then the projection map in the above proposition is not injective.
3.4
Cohomological
mod ` Forms Over K
For the purposes of this thesis, we will define modular forms over an imaginary quadratic field K cohomologically. Let OK denote the ring of integers of K. Analogous to the classical case, we define
32
the group Γ(n) for an ideal n ⊂ OK by (Ã Γ(n) =
a b
!
à ∈ GL2 (OK ) |
c d
a b
!
à ≡
c d
1 0
!
) mod n .
0 1
A congruence subgroup Γ is a subgroup of GL2 (OK ) which contains Γ(n) for some ideal n. The following two congruence subgroups are of particular importance: (Ã Γ0 (n) =
a b
!
à ∈ GL2 (OK ) |
c d
a b
!
à ≡
c d
∗ ∗
!
) mod n
0 ∗
and (Ã Γ1 (n) =
a b c d
!
à ∈ GL2 (OK ) |
a b c d
!
à ≡
∗ ∗ 0 1
!
) mod n .
Our definition of modular forms involves certain operators Tp called Hecke operators, which we will define in Section 5.5. Definition 3.4.1. We define a cohomological mod ` form of level n and Serre weight V to be a non-trivial cohomology class v ∈ H 2 (Γ, V ) which is a simultaneous eigenvector for the Hecke operators Tp for all primes p such that p - `n. Here, Γ is a congruence subgroup of level n.
3.5
Serre’s Conjecture for Imaginary Quadratic Fields
We need to define what it means for a representation ρ to be modular of weight ¯ ` -representation of G = V . Here, V is a Serre weight, i.e., V is an irreducible F GL2 (O/`O). We have seen that such representations are of the form V =
O p|`
33
Vp
where Vp =
O¡
¢ ¯ `. detaτ ⊗kp Symbτ −1 kp2 ⊗τ F
τ ∈Sp
To a modular form f we associate a homomorphism, which we will again denote ¯ ` [Tr : r - `n], where Tr by f , as follows. We first define the Hecke algebra, T = F is the Hecke operator for r prime. Then we can consider a modular form f to be a homomorphism ¯ ` [ar : r - `n] f :T→F Tr 7→ ar , where ar is the eigenvalue of the form f for the Hecke operator Tr . ¯ ` ), we can associate a maximal ideal To a Galois representation ρ : GK → GL2 (F in the Hecke algebra T as follows. From ρ, we get a map ¯` T→F Tr 7→ tr(ρ(Frobr )). We then get an exact sequence ¯ `, 0 → mρ → T → F with mρ = hTr − tr(ρ(Frobr ))i ⊂ T the maximal ideal we associate to ρ. We also define MΓ,V = H 2 (Γ, V ). ¯ ` ) be a continuous, irreducible representation Definition 3.5.1. Let ρ : GK → GL2 (F and V a Serre weight. We say that ρ is modular of weight V if MΓ,V [mρ ] 6= 0. Note that this is equivalent to requiring that there be a non-zero modular form ¯ ` for all f ∈ MΓ,V such that the eigenvalue ar of Tr (f ) is equal to tr(ρ(Frobr )) in F primes r - `n. The main question concerning this thesis is the following. Question 3.5.2. Let K be an imaginary quadratic field and suppose ¯ `) ρ : GK → GL2 (F 34
is a continuous, irreducible representation. Is it true that ρ is modular of weight V for every Serre weight V in the BDJ conjectural weight set W (ρ)? Remark 3.5.3. In Serre’s original conjecture, he requires that the representation ρ be odd. For a representation of GK where K is an imaginary quadratic field, there is no odd/even distinction.
35
Examples of Galois Representations In this chapter, we compute examples of Galois representations. The examples we compute come from three sources: elliptic curves, class field theory and representations arising from polynomials. Much of the data in this chapter was computed with a variety of mathematical software systems, including KANT/KASH [DF+ 97], Magma [BCP97], PARI/GP [The05] and Sage [Ste].
4.1
Examples from Elliptic Curves
Fix a prime ` and let E be an elliptic curve over K = Q(i) which is supersingular at ` and which has good reduction at `. To such an elliptic curve E one can associate a mod ` Galois representation ρE such that 1. the level of ρE is equal to the conductor of the elliptic curve E; 2. the character of ρE is trivial; and 3. the traces ap of ρE (Frobp ) are given by the sequence {ap }E associated to the elliptic curve, reduced mod `. Furthermore, with the above assumptions, we know the local behaviour of ρE at `. In particular, we will look at the case ` = 7. Letting I7 denote the inertia group at 7, we have
à ρ E |I 7 ∼
ω
0
0 ω7
! ,
where ω denotes the level 2 fundamental character. Using the BDJ recipe, we have χ1 = ω 1 χ2 = ω 7 ,
36
so n1 = 1 and n2 = 7. The inertia degree for 7 in K = Q(i) is f = 2, so we will have a ∈ Z/(`f − 1)Z = Z/48Z ~b = (b0 , b1 ) and B ⊂ {0, 1}. We need to find triples (a, ~b, B) such that n1 ≡ na,~b,B = a +
X
7i bi
mod 48,
7i bi
mod 48.
and
i∈B
n2 ≡ na,~b,B¯ = a +
X ¯ i∈B
For each of the four subsets B, we get one solution (a, ~b). We then compute ~a = (a0 , a1 ) by computing a ≡ a0 + 7a1 mod 48. We list the predicted weights in Table 4.2. Table 4.1: Weights for elliptic curves mod 7 ` char ~a = (a0 , a1 ) ~b = (b0 , b1 ) 7 7 7 7
1 1 1 1
(0, 0) (0, 0) (1, 0) (0, 1)
(1, 1) (7, 7) (5, 7) (7, 5)
The examples we compute in this section come from elliptic curves over Q(i) computed by Cremona in [Cre84]. Cremona computed modular forms (with coefficients in Q, i.e. with trivial weight) corresponding to these elliptic curves, so we already know that the representations associated to these elliptic curves are modular. Here we compute all the weights in which we expect to find modular forms giving rise to these representations. By computing the modular forms for those weights and levels, we provide computational evidence for the BDJ conjecture in this setting.
37
In his tables, Cremona lists elliptic curves by giving their ai coefficients for the Weierstrass equation E : y 2 + a1 xy + a3 y = x3 + a2 x2 + a4 x + a6 . In order to determine whether these elliptic curves are supersingular and have good reduction at `, we first complete the square to get an equation of the form E : y 2 = 4x3 + b2 x2 + 2b4 x + b6 where b2 = a21 + 4a2 b4 = 2a4 + a1 a3 b6 = a23 + 4a6 . We will also need b8 = a21 a6 + 4a2 a6 − a1 a3 a4 + a2 a23 − a24 , and ∆ = −b22 b8 − 8b34 − 27b26 + 9b2 b4 b6 . This last quantity ∆ is called the discriminant of the Weierstrass equation. Considering ∆ modulo ` tells us whether E has good reduction modulo `, i.e., whether the reduction of E modulo ` is non-singular. To test whether E is supersingular at `, we use the following theorem. (See, e.g., [Sil86, p. 140].) Theorem 4.1.1. Suppose K is a finite field of characteristic ` 6= 2. Let E be an elliptic curve over K given by Weierstrass equation E : y 2 = f (x), ¯ The curve E is superwhere f (x) ∈ K[x] has degree 3 and has distinct roots in K. singular if and only if the x`−1 term in f (x)(`−1)/2 has coefficient equal to zero.
38
Of the curves listed in Cremona’s table, we found several which are both supersingular and have good reduction at ` = 7. We list the ai coefficients for these curves in the following table, along with their conductors f and the norms of the conductors N f. We indicate those curves whose sequences {ap } suggest that the corresponding modular form is actually Galois over Q (judging by whether ap equals a¯p for split primes pOK = pp¯). We are, of course, more interested in those which are not Galois over Q. In the rightmost column, we indicate whether the corresponding modular form was found for all of the predicted weights (listed above in Table 4.1). Table 4.2: Elliptic curves over Q(i) mod 7 with several levels a1 1+i i 1 i
a2 0 1−i 1 −1
a3 1+i i 1 0
a4 a6 1−i 0 −i 0 0 0 2−i i
f Nf (6 + 6i) 72 (9 + 7i) 130 (15) 225 (19 + 9i) 442
Gal/Q X X
f X X X X
In Section 6.3, we give the {ap }E sequence for some small primes p and the {ap } coefficients computed for the corresponding modular forms.
4.2
Examples from Polynomials
In this section we will examine examples of Galois representations arising from polynomials. We will determine the level, character and predicted weights for each representation.
4.2.1 Techniques & Background Here we will review some background, stating basic theorems and describing general techniques, which will be used in the subsequent sections to analyze the various examples of Galois representations. Let p(x) be an irreducible polynomial in Q[x], and denote by L the Galois closure of the field generated by p(x) over Q. We will look at 2-dimensional 39
mod `
¯ representations of the Galois group Gal(L/Q). Taking the image of Gal(Q/L) to be trivial, we get a representation ¯ ` ), ρQ : GQ → GL2 (F which factors through Gal(L/Q). We are interested in representations of GK for imaginary quadratic fields K, so we take the restriction to K of the above representation, giving us ¯ ` ), ρK : GK → GL2 (F which factors through Gal(M/K), where M is the composite field LK. If we start with a representation ρQ over Q which is odd, i.e. det ρ(σ∞ ) = −1, then we already know ρQ is modular by Serre’s conjecture. Instead, we will start with even representations ρQ , and look at the base change to K. Once we have a representation ¯ ` ), ρK : GK → GL2 (F we need to compute the level, weight and character of ρK in order to figure out where to look for corresponding modular forms. We will of course also have to compute the coefficients {ap } for primes p. To compute the level of ρK , we will look at all primes dividing the discriminant of the polynomial, except `. For each such prime p, we compute n(p, ρ|Q ). In the examples we compute we will have p unramified in K for all primes p dividing the level N of the representation ρ|Q . When this is the case, we have n(p, ρ|K ) = n(p, ρ|Q ) for primes p lying above p. The corresponding level to check will be n=
Y
pn(p,ρ) .
As discussed earlier, we know that for any prime p which is tamely ramified in L, we have n(p, ρ) = dim(V /V0 ), where V0 is the subspace of V fixed by the inertia group IP for some prime P of L lying above p. We often make use of the basic fact that |IP |
40
is equal to the ramification index eP/p . If p is wildly ramified in L, the determination of n(p, ρ) requires further analysis. To determine the possible Serre weights V~a,~b , we consider the primes p lying above `. For each such prime p, we need to consider the local behaviour of ρK at p. We then apply the weight recipe in [BDJ] to compute the possible weights Vp . There are two cases: when the restriction of ρK to the decomposition group Dp is irreducible and when it is reducible. In either case, we will examine the restriction of ρK to the inertia group Ip . We make use of the combinatorial analysis in [BDJ], which provides formulas for the number of different weights Vp for each prime p above `, to check that we have found all possible weights. In those cases where we compute the base change from a representation over Q, we make use of some basic facts regarding base change to compute the coefficients {ap }. Recall that we want to compute the restriction of the representation ρ to the imaginary quadratic field K = Q(i), ¯ ` ). ρK = ρ|K : GK → GL2 (F Let M = KL be the composite of K and L. Since both K and L are Galois over Q, we know that M is Galois over Q and M is Galois over K. In our examples, we will consider totally real extensions L, so we have K ∩ L = Q, and thus G = Gal(M/K) ∼ = Gal(L/Q) and
Gal(M/Q) ∼ = Gal(K/Q) × Gal(L/Q) σ 7→ (σ|K , σ|L ).
Suppose p is a rational prime which is unramified in M , and let P be a prime of M lying above p. Let pK = P ∩ K and, likewise, pL = P ∩ L. We need to compute apK for M over K from ap for L over Q. We examine the two cases: p splits in K and p is inert in K. Case 1 (p splits in K): We have p = pK p¯K . Suppose for some prime P of M
41
lying over p, we have pK = P ∩ K. The inertia degree f (pK /p) is equal to 1. Thus we have
·
¸ µ· ¸ · ¸¶ M/K K/Q L/Q = × . P pK pL
The decomposition group DpK is trivial. Then, since FrobpK ∈ DpK , we must have ·
We now have
¸ K/Q = FrobpK = 1. pK
µ· ρK
M/K P
µ·
¸¶ =ρ
L/Q pL
¸¶
and so, of course, the traces will be the same as well, i.e., apK = a¯pK = ap . Case 2 (p is inert in K): We have p = p2K . The inertia degree f (pK /p) is equal to 2. Thus we have
and so
·
µ· ρK
¸ µ· ¸ · ¸¶2 M/K K/Q L/Q = × P pK pL ÷ ¸2 · ¸2 ! L/Q K/Q × = pK pL à · ¸2 ! L/Q = id × , pL
M/K P
÷
¸¶ =ρ
L/Q pL
¸2 !
µ· 2
=ρ
L/Q pL
¸¶ .
We can use this relationship to compute ap from ap . Alternatively, we could use the following method. Denote by ap the trace of ρQ (Frobp ) and denote by bp the trace of ρK (FrobpK ). ¯ ` with αβ = det(ρQ (Frobp )). Then we have Suppose ap = α + β, for some α, β ∈ F bp = α 2 + β 2 = (α + β)2 − 2 det(ρQ (Frobp )) = a2p − 2 det(ρQ (Frobp )). We want to compute the sequence {ap } associated to the representation ρ, i.e. 42
we want to compute tr(ρ(Frobp )) for each prime p of OK where p is unramified in ¯K the residue field of M = LK. Let P be a prime of M lying over p. Denote by O OK and likewise for M . Since p is unramified in M , the Frobenius automorphism Frobp is uniquely defined ¯M /O ¯K ) of the residue ([Jan96, p 125]). Also, Frobp generates the Galois group Gal(O field extension corresponding to the field extension M over K. We know that ¯M /O ¯K ) ∼ Gal(O = Gal(Fqf /Fq ), ¯K | and f = f (P/p) is the inertia degree of P over p. Thus the order of where q = |O the element Frobp is equal to the inertia degree f . When the image ρ(Gal(M/K)) is isomorphic to Gal(M/K), we know that the order of ρ(Frobp ) is equal to the order of the Frobp . Often, the order of ρ(Frobp ) is sufficient to determine the trace of ρ(Frobp ).
4.2.2 Dihedral Group D4 In this section, we compute an example with dihedral group D4 , which appears in Ash, Doud and Pollack [ADP02]. Example 4.2.1. Representation
mod 5 with level n = 29
The totally real polynomial x4 − x3 − 3x2 + x + 1,
disc = 52 × 29,
has Galois closure L defined by the polynomial x8 − 6x7 − 13x6 + 102x5 + 27x4 − 438x3 + 25x2 + 252x + 49. The Galois group G = Gal(L/Q) is isomorphic to the dihedral group D4 . We will take ` = 5. There is one irreducible 2-dimensional mod 5 representation of D4 . Since the extension L over Q is totally real, this representation is necessarily even and hence not covered by Serre’s conjecture. (However, since it has solvable image, it is known
43
to be modular by proven cases of Artin’s conjecture.) We consider the base change ρK from Q to K = Q(i) of this representation. In the following table we give, for each conjugacy class in the image of ρ, a representative element along with its order, length (of conjugacy class), determinant, ap = trace, and bp = tr2 − 2 det. For a split prime pOK = pp¯ we have tr(ρ(Frobp )) = ap = a¯p and for an inert prime pOK = p we have tr(ρ(Frobp )) = bp . Ã Ã Ã Ã Ã
rep 1 0 0 1 4 0 0 4 0 1 1 0 4 0 0 1 0 4 1 0
!
order
length
det ap
bp
1
1
1
2
2
2
1
1
3
2
2
2
4
0
2
2
2
4
0
2
4
2
1
0
3
! ! ! !
From the above table, we see that det(ρ) is a quadratic character, and so we get a nontrivial character for ρ, which is a quadratic character on (OK /29OK )∗ . We will denote this character by ε29 . To compute the level, we find the ramification index at p = 29 is e29 = 2. Since p = 29 is tamely ramified, we have n(29, ρ) = dim(V /V0 ). The inertia group I29 has order 2 and is not central. The representation ρ restricted to such a subgroup of D4 fixes a space of dimension 1, so n(29, ρ) = 1 and the level of this representation is n = 29. We use the BDJ recipe to compute the weights for ` = 5. We compute the ramification index e5 = 2 and the inertia degree f5 = 2. Furthermore, we compute that ρ|I5 fixes a one dimensional subspace and so is of the form à ρ|I5 ∼
ω2 0 0
44
1
! ,
where ω is the mod 5 cyclotomic character. We use the reducible case of BDJ, and we have χ1 = ω 2 χ2 = ω 0 , so n1 = 2 and n2 = 0. The inertia degree for 5 in K = Q(i) is f = 1, so we will have a ∈ Z/(`f − 1)Z = Z/4Z ~b = (b0 ) and B ⊂ {0}. We need to find triples (a, b0 , B) such that n1 ≡ na,b0 ,B = a +
X
5i bi
mod 4,
5i bi
mod 4.
and
i∈B
n2 ≡ na,b0 ,B¯ = a +
X ¯ i∈B
For each of the two subsets B, we get one solution (a, b0 ). In particular, for B = {0}, we get a = 0, b0 = 2 and for B = ∅, we get a = 2, b0 = 2. The analysis is the same regardless of which prime we choose above ` = 5, and so, for K = Q(i), we combine the results to get the various possible weights, which we list in Table 4.3. Table 4.3: D4 representation mod 5 with level n = 29 ` level char ~a = (a0 , a1 ) ~b = (b0 , b1 ) f 5 29 ε29 (0, 0) (2, 2) X 5 29 ε29 (0, 2) (2, 2) 5 29 ε29 (2, 0) (2, 2) 5 29 ε29 (2, 2) (2, 2) X
For two of the four weights in the above table, we did not find the form as expected. There was a form found in those weights that looks as though it might be a twist of the expected form. Further investigation is required to determine what is happening here. In Section 6.4.1 we provide a table with the orders of Frobp along with the coeffi45
cients ap of the corresponding system of eigenvalues for some small primes p for this representation.
4.2.3 Alternating Group A4 Example 4.2.2. Representation
mod 3 with level n = 61
In [Fig99, p 117], Figueiredo gives three examples of A4 representations. The first of these examples comes from the polynomial x4 − 7x2 − 3x + 1
disc = 32 × 612 .
Figueiredo considers the mod 3 representation arising from this polynomial using the isomorphism A4 ∼ = PSL2 (F3 ). He shows that there must be a lift of this representation ¯ 3 ). I will instead compute representations directly from the Aˆ4 extension, to GL2 (F where Aˆ4 is a double cover of A4 , isomorphic to SL2 (F3 ). From the database of Kl¨ uners and Malle [KM], we find the polynomial giving the Aˆ4 extension of Figueiredo’s polynomial: x8 + 3x7 − 11x6 − 9x5 + 21x4 + 9x3 − 11x2 − 3x + 1 which has Galois closure given by the following degree 24 polynomial: x24 − 9x23 − 206x22 + 1680x21 + 16053x20 − 112863x19 − 585638x18 + 3495552x17 + 9763561x16 − 57615780x15 − 68523951x14 + 519956256x13 + 95882805x12 − 2594932704x11 + 1213595253x10 + 6932798424x9 − 6682968027x8 − 8573027148x7 + 13459018536x6 + 1679228685x5 − 10467091917x4 + 4152416400x3 + 1245998376x2 − 1023909417x + 155570139 We denote this Galois closure by L.
46
We take ` = 3. There is only one irreducible 2-dimensional mod 3 representation ¯ 3) ρ : GK → GL2 (F factoring through Gal(LK/K). We get this representation by taking the base change to K = Q(i) from the representation ρQ of GQ , which we get simply by restricting GQ to Gal(L/Q) and then applying the following isomorphisms and inclusion: Gal(L/Q) ∼ = Aˆ4 ∼ = SL2 (F3 ) ,→ GL2 (F3 ). For the level, we need only compute n(61, ρ). We have the ramification index e61 = 3 in L, and so 61 is tamely ramified and n(61, ρ) = dim(V /V0 ). We check that ρ restricted to the subgroup of order 3 fixes a 1-dimensional subspace of V , so n(61, ρ) = 1 and the level of ρ is n = 61. Since the image of ρ is SL2 (F3 ), we see that det(ρ) is trivial, and so the character of ρ is trivial as well. To compute the weights, we look at the representation locally at ` = 3 and apply the BDJ recipe. We compute the ramification index e3 = 4 and the inertia degree f3 = 2, and we note that the inertia degree of 3 in K = Q(i) is f = 2. The decomposition group at 3 for L over Q has order 8, and the only order 8 subgroup is the quaternion group Q8 . If we consider the restriction of ρQ to the decomposition group, we would be in the irreducible case of BDJ. However, we want to consider the restriction of ρQ first to K = Q(i), which we denote by ρ, and then to the decomposition group at 3. Then we have ρ restricted to D3 is reducible, and we can write
à ρ|D3 ∼
ω (`
2 −1)/4
and so we have χ1 = ω (`
0 ω −(`
0 2 −1)/4
2 −1)/4
= ω2
2 −1)/4
χ2 = ω −(`
!
= ω −2 .
Thus n1 = 2 and n2 = 6. (Recall that nν ∈ Z/(`f − 1)Z = Z/8Z.) We need to find 47
triples (a, ~b, B) with a ∈ Z/8Z B ⊂ {0, . . . , f − 1} and ~b = (b0 , b1 ), with 1 ≤ bi ≤ ` = 3 such that n1 ≡ na,~b,B = a +
X
3i bi
mod 8,
3i bi
mod 8.
and
i∈B
n2 ≡ na,~b,B¯ = a +
X ¯ i∈B
For two of the four subsets B (namely, B = {0} and B = {1}) we get one solution (a, ~b), while for the other two (B = {0, 1} and B = ∅) we get two solutions (a, ~b). We then compute ~a = (a0 , a1 ) by writing a = a0 + 3a1 mod 8. In Table 4.4 we list the possible weights. Table 4.4: A4 representation mod 3 with level n = 61 ` level B ~a = (a0 , a1 ) ~b = (b0 , b1 ) f 3 61 {0, 1} (0, 2) (1, 1) X 3 61 {0, 1} (0, 2) (3, 3) X 3 61 {1} (1, 1) (2, 2) X 3 61 {0} (0, 0) (2, 2) X 3 61 ∅ (2, 0) (1, 1) X 3 61 ∅ (2, 0) (3, 3) X
We need to compute the coefficients {ap }. When p splits in K, say p · OK = pp¯, we have ap = a¯p = ap . The image of our representation is SL2 (F3 ), so we have det = 1. Therefore if p = p is inert in K, we get bp = tr(ρ(Frobp ))2 − 2 · det(ρ(Frobp )) = b2p − 2 ≡ b2p + 1
mod 3.
In the following table we give representatives for each conjugacy class in SL2 (F3 ), 48
the order of those elements, the trace (equal to ap when p splits in K) and bp when p is inert in K. Rep
à à à à à à Ã
1 0
!
0 1 −1
0
0
−1
0
1
−1 −1 0 −1 1 −1 0 −1 1
0
0 −1 1 0
1 1
−1 1
Order
tr = ap
bp
1
−1
−1
2
1
−1
3
−1
−1
3
−1
−1
4
0
1
6
1
−1
6
1
−1
! ! ! ! ! !
In Section 6.4.2 we provide a table with the orders of Frobp along with the coefficients of the corresponding systems of eigenvalues for some small primes p for this representation. Example 4.2.3. Representation
mod 3, level n = 79
Figueiredo’s second example in [Fig99] comes from the polynomial P = x4 − x3 − 24x2 + x + 11
disc = 34 × 792 .
This is a totally real A4 extension of Q. It is a subfield of an Aˆ4 extension generated by the following polynomial x8 − 17x6 + 60x4 − 29x2 + 1,
49
which has as its Galois closure the field defined by the following degree 24 polynomial x24 − 255x22 + 27807x20 − 1706560x18 + 65353749x16 − 1637285385x14 + 27343876078x12 − 303898414365x10 + 2196602852661x8 − 9790619278320x6 + 24093731990727x4 − 25093789540875x2 + 2600398130625. Let L denote the field generated by this degree 24 extension. We take ` = 3, so we will be considering mod 3 representations of Gal(L/Q) ∼ = Aˆ4 ∼ = SL2 (F3 ). There is only one such irreducible degree 2 representation. We need to compute the level of this representation. Since we are considering this representation mod 3 and the only primes ramifying in this extension are 3 and 79, we need only consider p = 79 for the level. The ramification at p = 79 is tame and so n(79, ρ) = dim(V /V0 ). We compute the ramification index e79 = 3 and denote the inertia group at 79 by I79 . We find that ρ|I79 fixes a 1-dimensional subspace of V , so n(79, ρ) = 1 and the level of this representation is n = 79. Since the image of this representation is isomorphic to SL2 (F3 ), the character will be trivial. We now use the BDJ recipe to compute the weights. For ` = 3 we find the ramification index e3 = 3 and the inertia degree f3 = 2. Globally, the decomposition group at 3 would then have order 6. However, as in the previous example, we consider the restriction of the representation to K = Q(i), and so the decomposition group D3 in this case will have order 3. We are thus in the reducible case of BDJ. We can write
à ρ|D3 ∼
1 ∗ 0 1
and so we have χ1 = ω 0 χ2 = ω 0 ,
50
! ,
with n1 = n2 = 0 (and so also n = 0). We need to find triples (a, ~b, B) with a ∈ Z/8Z, ~b = (b0 , b1 ) with 1 ≤ bi ≤ ` = 3 and B ⊂ {0, 1} such that 0 ≡ na,~b,B = a +
X
3i bi
mod 8,
3i bi
mod 8.
and
i∈B
0 ≡ na,~b,B¯ = a +
X ¯ i∈B
For two of the four subsets B (namely, B = {0, 1} and B = ∅) we get one solution ~b (but two solutions for (~a, ~b)), while for the other two (B = {0} and B = {1}) we get two solutions (a, ~b) each. However there is some symmetry here: we get the same solutions for B = {0} as for B = {1}, and we get the same solutions for B = ∅ as for B = {0, 1}. This symmetry is expected by Proposition 3.3.6, since n = 0. We compute ~a = (a0 , a1 ) by writing a = a0 + 3a1 mod 8. In Table 4.5 we list the possible weights. Table 4.5: A4 representation mod 3 with level n = 79 ` level char B ~a = (a0 , a1 ) ~b = (b0 , b1 ) f 3 3 3
79 79 79
1 1 1
{0, 1}, ∅ {1}, {0} {1}, {0}
(0, 0) (1, 2) (2, 1)
(2, 2) (1, 3) (3, 1)
X
The weights in Table 4.5 are the weights in W 0 (χ1 , χ2 ) in the BDJ notation for the reducible case. Recall that, for these weights to be in the predicted weight set Wp (ρ), they must satisfy the additional condition that the corresponding cohomology ¯ ` (χ1 χ−1 )). In this case, one class cρ be contained in a certain subspace Lα ⊂ H 1 (Kp , F 2
can show that this condition is not met for the latter weights (those with ~b = (1, 3) and ~b = (3, 1)). Thus we only expect this form to show up in the former weight (with ~b = (2, 2)) and, as indicated in the final column, that is indeed what we found. In Section 6.4.2 we provide a table with the orders of Frobp along with the coef51
ficients of the corresponding systems of eigenvalues for some small primes p for this representation.
4.3
Examples from Class Field Theory
4.3.1 Dihedral Group D3 For this section, we assume ` 6= 3. Write the elements of D3 as rotations rk and ¯ ∗ , we get one irreducible reflections srk for 0 ≤ k ≤ 2. If w is a third root of unity in F `
2-dimensional representation ρ of D3 . We can write the action of ρ on the elements of D3 as follows (see, e.g., [Ser77, p.36]) Ã ρ(rk ) =
wk 0
!
0 w
à ρ(srk ) =
,
−k
0
w−k
wk
0
! .
In the following table, we give the trace, determinant and order of the images of ρ on the elements of D3 . r0
r1
r2
sr0
sr1
sr2
trace
2
−1 −1
0
0
0
det
1
1
1
−1
−1
−1
order
1
3
3
2
2
2
To compute the coefficients bp in the base change to K = Q(i) when p is inert, we use the formula bp = tr(ρQ (Frobp ))2 − 2 · det(ρQ (Frobp )), where p is the rational prime below p. We get the following values for bp when p is inert in Q(i).
bp
r0
r1
2
−1 −1
Example 4.3.1. Representations
r2
sr0
sr1
sr2
2
2
2
mod 5 and
13 + 28i and n = 8 + 35i 52
mod 7 with levels n = 8 + 17i, n =
The following examples are not arising as base change of even representations over Q, but are representations directly over K = Q(i). We get these examples by considering quadratic extensions of Q(i) which are ramified only at a single prime p, split over Q, and which have class group isomorphic to the cyclic group of order 3. I would like to thank Gabor Wiese for showing me this source of examples. At the time, we had been considering these examples
mod 2, but as 2 is ramified in
Q(i), it is not covered by the BDJ recipe. Schein’s extension of the BDJ recipe to the ramified case ([Sch]) does not cover ` = 2. In the following we will consider these representations mod 5 and mod 7. This allows us to see the different behaviour in the weights modulo an inert prime as compared to a split prime. The representation ρ will factor through an extension L of K = Q(i) where G = Gal(L/K) is isomorphic to D3 . In all cases the representation will have a ¯ ∗. quadratic character ε : (OK /p)∗ → F `
For the level, we need only consider this single ramified prime p, which will have ramification index ep = 2 in the extension L over K. The representation ρ restricted to the order 2 subgroup of D3 fixes a one-dimensional subspace, so that the level in each case will be precisely equal to this prime, i.e., n = p. In all three examples, the primes above 5 and 7 in Q(i) will be unramified in the dihedral extension L over K. Thus, in all cases, the representation ρ restricted to the inertia group I at ` will be trivial and so we have χ1 = ω 0 χ2 = ω 0 , and n = n1 = n2 = 0. The difference in the weight computations arises from the prime ` being split or inert. Note that for both cases n = 0 implies, according to Proposition 3.3.6, that we will have multiple subsets B with the same (a, ~b).
53
For ` = 5 (exemplifying the split case), we need to find triples (a, b0 , B) with a ∈ Z/4Z, 1 ≤ b0 ≤ 5 and B ⊂ {0} such that
X
0 ≡ na,b0 ,B = a +
5i bi
mod 4,
5i bi
mod 4.
and
i∈B
X
0 ≡ na,b0 ,B¯ = a +
¯ i∈B
If we take either B = {0} or B = ∅, we have the congruences 0 ≡ a + b0 0≡a
mod 4,
and
mod 4.
Thus a is 0 and b0 = 4. For ` = 7 (exemplifying the inert case), we look for triples (a, ~b, B) with a ∈ Z/48Z, ~b = (b0 , b1 ) with 1 ≤ bi ≤ 7 and B ⊂ {0, 1} such that 0 ≡ na,b0 ,B = a +
X
7i bi
mod 48,
7i bi
mod 48.
and
i∈B
0 ≡ na,b0 ,B¯ = a +
X ¯ i∈B
We again have some symmetry: the cases B = {0, 1} and B = ∅ will yield the same system of congruences and the cases B = {1} and B = {0} will yield the same system of congruences. For the former, we have 0 ≡ a + b0 + 7b1
mod 48,
0 ≡ a mod 48, 54
and
giving ~b = (6, 6) and a = 0 so ~a = (0, 0). In the latter case, we have 0 ≡ a + b0 0 ≡ a + 7b1
mod 48,
and
mod 48,
giving ~a = (5, 6) with ~b = (1, 7) and ~a = (6, 5) with ~b = (7, 1). The primes for which we found such dihedral extensions are n = (8 + 17i) (lying over p = 353), n = (13 + 28i) (lying over p = 953) and n = (8 + 35i) (lying over p = 1289). In Table 4.6, we list all the combinations of weights, level, character and prime ` for the above examples, and indicate in the final column (under f ) whether a corresponding modular form was found. Table 4.6: D3 representations mod 5 and mod 7 with several levels ` 5 7 7 7 5 7 7 7 5 7 7 7
level 8 + 17i 8 + 17i 8 + 17i 8 + 17i 13 + 28i 13 + 28i 13 + 28i 13 + 28i 8 + 35i 8 + 35i 8 + 35i 8 + 35i
char ~a = (a0 , a1 ) ~b = (b0 , b1 ) ε8+17i (0, 0) (4, 4) ε8+17i (0, 0) (6, 6) ε8+17i (5, 6) (1, 7) ε8+17i (6, 5) (7, 1) ε13+28i (0, 0) (4, 4) ε13+28i (0, 0) (6, 6) ε13+28i (5, 6) (1, 7) ε13+28i (6, 5) (7, 1) ε8+35i (0, 0) (4, 4) ε8+35i (0, 0) (6, 6) ε8+35i (5, 6) (1, 7) ε8+35i (6, 5) (7, 1)
f X X X X X X X X X X X X
In Section 6.5.1 we provide tables with the orders of Frobp along with the coefficients ap of the corresponding system of eigenvalues for some small primes p for each of the three levels n above. 55
4.3.2 Dihedral Group D5 Example 4.3.2. Representation
mod 11 with level n = 19 + 20i
In this section we look at another example which is a representation directly over K = Q(i), not a base change from a representation over Q. This one was constructed in the same way as the D3 examples of this sort in Section 4.3.1. For the prime 19 + 20i (lying over p = 761), we get a quadratic extension of Q(i) which is ramified only at 19 + 20i and which has class group isomorphic to the cyclic group of order 5. Thus we get an extension L of K = Q(i) such that the Galois group G = Gal(L/K) is isomorphic to D5 . We will consider this representation modulo ` = 11, for which prime we have an irreducible 2-dimensional representation of D5 . The image of det(ρ) is ±1, so the ¯ ∗. character of ρ will be a quadratic character ε19+20i : (OK /(19 + 20i)OK )∗ → F `
In the following table we give, for each conjugacy class in the image of ρ, a representative element along with its order, length (of conjugacy class), determinant and trace. Ã Ã Ã Ã
rep 1 0 0 1 0 1 1 0 4 0 0 3 5 0 0 9
!
order
length
det
trace
1
1
1
2
2
5
10
0
5
2
1
7
5
2
1
3
! ! !
The ramification index of p = 19 + 20i in L over K is ep = 2. The fixed space of ρ restricted to the order 2 subgroup of D5 has dimension 1 so the level of ρ is n = 19 + 20i. To compute the weights, we apply the BDJ recipe in the reducible case. The prime above 11 is unramified in the extension L over K, so the restriction of ρ to the
56
inertia group I for ` = 11 is trivial. Thus we have χ1 = ω 0 χ2 = ω 0 , so n1 = n2 = 0. Note that n = 0 implies, according to Proposition 3.3.6, that we will have multiple subsets B with the same (a, ~b). We look for triples (a, ~b, B) with a ∈ Z/120Z, ~b = (b0 , b1 ) with 1 ≤ bi ≤ 11 and B ⊂ {0, 1} such that 0 ≡ na,b0 ,B = a +
X
11i bi
mod 120,
11i bi
mod 120.
and
i∈B
0 ≡ na,b0 ,B¯ = a +
X ¯ i∈B
For the subsets B = {0, 1} and B = ∅ we get the same system of congruences, namely: 0 ≡ a + b0 + 11b1
mod 120,
and
0 ≡ a mod 120, giving ~b = (10, 10) and a = 0 so ~a = (0, 0). For the subsets B = {1} and B = {0} we get the same system of congruences, namely: 0 ≡ a + b0 0 ≡ a + 11b1
mod 120,
and
mod 120,
giving ~a = (9, 10) with ~b = (1, 11) and ~a = (10, 9) with ~b = (11, 1). In Table 4.7, we list the possible weights with the level and character for this example and indicate in the final column whether a corresponding form was found. The case for which it has not yet been found has not been checked due to size limitations in the computations.
57
Table 4.7: D5 representation mod 11 with level n = 19 + 20i ` level char ~a = (a0 , a1 ) ~b = (b0 , b1 ) f 11 11 11
19 + 20i ε19+20i 19 + 20i ε19+20i 19 + 20i ε19+20i
(0, 0) (9, 10) (10, 9)
(10, 10) (1, 11) (11, 1)
X X
In Section 6.5.2 we provide a table with the orders of Frobp along with the coefficients of the corresponding systems of eigenvalues for some small primes p for this representation.
58
Computing Modular Forms over Q(i) 5.1
Borel-Serre Duality
Recall that the space we want to compute is H 2 (Γ, V ) for some congruence subgroup Γ and some Serre weight V . Instead of computing this cohomology group directly, we use Borel-Serre duality to compute a homology group with coefficients in the Steinberg module. The homology group is computationally friendly because we can express the Steinberg module (and a free resolution of it) in terms of modular symbols. Let K be a number field with ring of integers O and let Γ be a subgroup of finite index in GL2 (O). We will assume that K has class number 1 so that O is a PID. Denote by ν the virtual cohomological dimension of Γ. Applying the formula in Ash [Ash94, p.330], we see that ν = 2r1 + 3r2 − 1, where r1 is the number of real and 2r2 the number of complex embeddings of K. Let R be a commutative ring with identity such that the torsion in Γ is invertible in R. We define the Steinberg module to be the R[Γ]-module H ν (Γ, R[Γ]) and denote it by St. Suppose M is an R[Γ]-module. Borel-Serre duality [BS73, p.482-483] gives an isomorphism ∼
H α (Γ, M ) −→ Hν−α (Γ, St ⊗ M ) for any integer α. For imaginary quadratic fields K, we have ν = 2. In our case, we want to compute H 2 (Γ, V ) for a Serre weight V , so via Borel-Serre duality, we will instead be computing H0 (Γ, St⊗V ). We will need to know how to compute the Steinberg module. Ash gives a description, which we recall here, of the Steinberg module in terms of “universal minimal modular symbols”. In [Ash94], Ash defines this space for arbitrary dimension n. Here, we restrict to the case n = 2. The space of universal minimal modular symbols is canonically isomorphic to the
59
Steinberg module as we have defined it above; from now on we will take the modular symbols description as our definition. Consider the set of formal R-linear sums of symbols [v] = [v1 , v2 ] where the vi are unimodular columns in O2 , i.e., vi = [a b]T with gcd(a, b) = 1. Mod out by the R-module generated by the following elements: 1. [v2 , v1 ] + [v1 , v2 ]; 2. [v] = [v1 , v2 ] whenever det(v) = 0; and 3. [v1 , v3 ] − [v1 , v2 ] − [v2 , v3 ], where the vi again run over all unimodular columns in O2 . This quotient module is now our definition of the Steinberg module St. We denote the image of [v] in St by [v]∗ . In order to compute the homology of Γ with coefficients in St ⊗ M , we will first give a free resolution of St · · · → S2 → S1 → S0 → St and then tensor with the R[Γ]-module M · · · → S2 ⊗ M → S1 ⊗ M → S0 ⊗ M → St ⊗ M. We will then take Γ coinvariants: · · · → (S2 ⊗ M )Γ → (S1 ⊗ M )Γ → (S0 ⊗ M )Γ → (St ⊗ M )Γ and, finally, take the homology of this chain complex. The first thing we need is an R[Γ]-free resolution of St. We now describe the resolution given by Ash in [Ash94], which is based on the resolution in Lee and Szczarba [LS76]. The R[Γ]-modules Sk of this resolution are described in a manner similar to that of the Steinberg module. Consider the set of formal R-linear sums of symbols [v] = [v1 , v2 , · · · , v2+k ], where each vi is a unimodular column in O2 . Mod out by the R[Γ]-module generated by the following elements: 60
1. [vσ(1) , . . . , vσ(2+k) ] − sgn(σ)[v1 , . . . , v2+k ]; and 2. [v] if v1 , . . . , v2+k are all contained in a hyperplane in K 2 . Here, the vi again run over all unimodular columns in O2 and σ runs over all permutations of {1, . . . , 2 + k}. We take this quotient module to be Sk in our free resolution of St: · · · → S2 → S1 → S0 → St and denote the image of [v] in Sk by [v] again. We define the boundary operator Sk → Sk−1 to be ∂[v] =
X
(−1)i [v1 , . . . , vˆi , . . . , v2+k ]
and the map S0 → St is simply [v] 7→ [v]∗ . Ash [Ash94, p.332] notes that the proof that the Sk chain complex is an R[Γ]-free resolution of St is similar to the proof given in [LS76]. We are interested in computing H 2 (Γ, M ), so (recalling that, for us, ν = 2) we apply the Borel-Serre isomorphism ∼
H 2 (Γ, M ) −→ H0 (Γ, St ⊗ M ) to see that we will instead be computing H0 (Γ, St ⊗ M ). Thus we will only be concerned with the following part of the chain complex: S1 −→ S0 −→ St. We tensor with M and then take coinvariants by Γ to get ∂
(S1 ⊗ M )Γ −→ (S0 ⊗ M )Γ −→ (St ⊗ M )Γ . We will compute the homology of degree 0 of this, i.e., H0 (Γ, St ⊗ M ) = ker ε/im∂, 61
where ∂
ε
(S1 ⊗ M )Γ −→ (S0 ⊗ M )Γ −→ 0. So we will compute (S0 ⊗ M )Γ /∂((S1 ⊗ M )Γ ), which amounts to computing (S0 ⊗M )Γ modulo the relations [v1 , v3 ]−[v1 , v2 ]−[v2 , v3 ], where the vi run over all unimodular columns in O2 . Note that, in our case, for both St and S0 , we are looking at a space of ordered pairs of unimodular columns in O2 , which is the same as paths between cusps (K ∪ {∞}) in the modular symbols methods of Cremona et al.
5.2
An Algebraic Proposition
The following proposition is analogous to Proposition 4.3 in the doctoral thesis of Martin [Mar01, p.69]. It will be used in Section 5.3 below to relate Manin symbols to modular symbols. First, we will need some notation. For now, let R be a ring and K = Q(i). Define the following matrices in GL2 (OK ): Ã J=
i 0 0 1
!
à S=
0 i
!
à T =
1 0
1 1
!
0 1
à T0 =
1 0
!
1 1
Proposition 5.2.1. Consider the following homomorphism of left R[GL2 (OK )]-modules Ψ : R[GL2 (OK )] −→ R[P1 (K)] X X uM [M ] 7−→ uM ([M (∞)] − [M (0)]). M
M
The kernel of Ψ is equal to the left R[GL2 (OK )] ideal J = h[I] − [T ] − [T 0 ], [I] + [S], [I] − [J]i. Proof. First, we will show J ⊆ ker(Ψ). For this we simply evaluate Ψ on [I]−[T ]−[T 0 ],
62
on [I] + [S] and on [I] − [J]. We have Ψ([I] − [T ] − [T 0 ]) = [I(∞)] − [I(0)] − [T (∞)] + [T (0)] − [T 0 (∞)] + [T 0 (0)] = [∞] − [0] − [∞] + [1] − [1] + [0] =0 and Ψ([I] + [S]) = [I(∞)] − [I(0)] + [S(∞)] − [S(0)] = [∞] − [0] + [0] − [∞] =0 and Ψ([I] − [J]) = [I(∞)] − [I(0)] − [J(∞)] + [J(0)] = [∞] − [0] − [∞] + [0] = 0. The other direction, proving that ker(Ψ) ⊆ J , requires more work. Let W = P 1 M uM [M ] be a non-zero element of ker(Ψ). Let L(W ) ⊆ P (K) be the union P P of supports of M uM [M (∞)] and M uM [M (0)]. Furthermore, we define L(W ) = αmax (|α|2 + |β|2 ) β
and
∈L(W )
¯½ ¾¯ ¯ α ¯ 2 2 m(W ) = ¯¯ ∈ L(W ) : |α| + |β| = L(W ) ¯¯ β
where we assume (α, β) = 1. We will use elements of the ideal J to write down a W 0 congruent to W modulo J , but such that L(W 0 ) ≤ L(W ). Futhermore, if L(W 0 ) = L(W ) then we will have m(W 0 ) < m(W ). Iterating this process, we will see that W ∈ J . Let
α β
∈ L(W ) be such that |α|2 + |β|2 = L(W ). Let δ, γ be elements of OK
such that αγ − βδ = 1 and such that |γ| ≤ |β| and |δ| ≤ |α|. Then the matrices of
63
GL2 (OK ) satisfying M (∞) = α/β are of the form à M=
im α in (δ + kα)
!
im β in (γ + kβ)
for k ∈ OK and m, n ∈ {0, 1, 2, 3}. As L(W ) = |α|2 + |β|2 , we see that for such a matrix M to be in the support of W , we must have |δ + kα|2 + |γ + kβ|2 ≤ |α|2 + |β|2 . We will show first that for this to be true, we must have |k| < 2. First, suppose |k| ≥ 2. Then |δ + kα|2 + |γ + kβ|2 ≥ (|kα| − |δ|)2 + (|kβ| − |γ|)2 ≥ (2|α| − |δ|)2 + (2|β| − |γ|)2 = (|α| + (|α| − |δ|))2 + (|β| + (|β| − |γ|))2 ≥ |α|2 + |β|2 . Note, furthermore, that equality is only possible if |α| = |δ| and |β| = |γ|, but this contradicts the fact that αγ − βδ = 1. Thus we must have |k| < 2. Since k ∈ OK , the only possibilities are 1. k = 0; ∗ 2. k ∈ OK ; or
3. |k|2 = 2, i.e., k ∈ {1 + i, 1 − i, −1 + i, −1 − i}. Aside: We show why |α| = |δ| and |β| = |γ| implies that M cannot have deter∗ minant in OK . We have det(M ) = im+n (αγ − βδ). Under our assumption, we have ∗ |αγ| = |βδ|, so we need to show that we cannot have x − y ∈ OK if |x| = |y|. Suppose
this is the case and let x = a + bi and y = c + di, so that x − y = (a − c) + (b − d)i. Then either a = c and b = d ± 1 or b = d and a = c ± 1. Without loss of generality,
64
assume a = c and b = d ± 1. Then |x|2 = |y|2 a2 + b2 = c2 + d2
⇒
(d ± 1)2 = d2 ,
⇒ giving a contradiction. Note that the action of J is Ã
a b
!
à J=
c d
a b
!Ã
c d
i 0
!
à =
0 1
ia b
! ,
ic d
so, with (possibly repeated) applications of I − J, we may assume the matrix M has the form
à M=
α in (δ + kα)
! .
β in (γ + kβ)
Also, the action of S is given by Ã
a b
!
c d
à S=
a b
!Ã
c d
0 i
!
à =
1 0
b ia d ic
! ,
so, with (possibly repeated) applications of I − J and I + S, we may further assume the matrix M has the form à Mk =
α δ + kα
! .
β γ + kβ
Using the same techniques, we may assume that the only matrices N in the support of W such that N (0) = α/β are those of the form à Nj =
jα − δ α jβ − γ β
! ,
where |j|2 ∈ {0, 1, 2} and |jα − δ|2 + |jβ − γ|2 ≤ |α|2 + |β|2 . Each of the Nj 65
can be replaced, using S and J, by −M−j as follows. Note that I + JS ∈ J since I + JS = (I − J) + J(I + S). Then Nj − Nj (I + JS) = −Nj JS = −M−j . Let W 0 denote the element W ∈ ker(Ψ) with the above modifications. So now all matrices M in the support of W 0 with M (∞) = α/β are of the form à Mk =
α δ + kα β γ + kβ
! ,
with |δ + kα|2 + |γ + kβ|2 ≤ |α|2 + |β|2 . Furthermore, we have no matrices N in the support of W 0 with N (0) = α/β. The strategy from here is as follows. We will first consider the case where |k|2 = 2, and we will replace such matrices M by two matrices M 0 and M 00 , where M 0 will have the same form as M above but with |k| = 1 and M 00 will be such that M 00 (∞) 6= α/β 6= M 00 (0). These matrices will also satisfy m(M ) = m(M 0 − M 00 ), so that the number of α/β ∈ L(W ) will not increase. The next step is to work with the case |k| = 1. We will replace each M of this sort again with two matrices M 0 and M 00 . One of these will be in the form of Mk but with k = 0. The other will again be such that M 00 (∞) 6= α/β 6= M 00 (0). As in the |k|2 = 2 case, these matrices will satisfy m(M ) = m(M 0 − M 00 ). After these steps, the only matrix M left in W 0 with M (∞) = α/β is M0 , and there are no matrices left in W 0 with M (0) = α/β. Since W 0 ∈ ker(Ψ), we must have the coefficient uM0 of M0 in W 0 equal to 0. We thus decrease m(W 0 ) by at least one.
66
To aid in our analysis, we write α = a1 + a2 i β = b1 + b2 i δ = c1 + c2 i γ = d1 + d2 i. To deal with the case |k|2 = 2, we start by proving the following claim. Claim 5.2.2. Suppose k ∈ OK is such that |k|2 = 2, i.e., k ∈ {1 + i, 1 − i, −1 + i, −1 − i} and write k = k1 + k2 i. If |δ + kα|2 + |γ + kβ|2 ≤ |α|2 + |β|2 , then either |δ + k1 α|2 + |γ + k1 β|2 or |δ + k2 iα|2 + |γ + k2 iβ|2 is strictly less than |α|2 + |β|2 . Proof. Consider k = 1 + i. Our assumption then becomes |c1 + c2 i + (1 + i)(a1 + a2 i)|2 + |d1 + d2 i + (1 + i)(b1 + b2 i)|2 = |(a1 − a2 + c1 ) + (a1 + a2 + c2 )i|2 + |(b1 − b2 + d1 ) + (b1 + b2 + d2 )i|2 = 2(a21 + a22 ) + 2a1 c1 − 2a2 c1 + 2a1 c2 + 2a2 c2 + c21 + c22 + 2(b21 + b22 ) + 2b1 d1 − 2b2 d1 + 2b1 d2 + 2b2 d2 + d21 + d22 ≤ a21 + a22 + b21 + b22 , i.e., a21 + a22 + 2a1 c1 − 2a2 c1 + 2a1 c2 + 2a2 c2 + c21 + c22 + b21 + b22 + 2b1 d1 − 2b2 d1 + 2b1 d2 + 2b2 d2 + d21 + d22 ≤ 0.
67
(5.1)
Suppose, by way of contradiction, that the claim is false for k = 1 + i. Then we have |α + δ|2 + |β + γ|2 = a21 + a22 + 2a1 c1 + 2a2 c2 + c21 + c22 + b21 + b22 + 2b1 d1 + 2b2 d2 + d21 + d22 ≥ a21 + a22 + b21 + b22 , that is, 2a1 c1 + 2a2 c2 + c21 + c22 + 2b1 d1 + 2b2 d2 + d21 + d22 ≥ 0.
(5.2)
We also have |iα + δ|2 + |iβ + γ|2 = a21 + a22 − 2a2 c1 + 2a1 c2 + c21 + c22 + b21 + b22 − 2b2 d1 + 2b1 d2 + d21 + d22 ≥ a21 + a22 + b21 + b22 , that is, −2a2 c1 + 2a1 c2 + c21 + c22 − 2b2 d1 + 2b1 d2 + d21 + d22 ≥ 0.
(5.3)
Adding together 5.2 and 5.3, we see that 2a1 c1 + 2a2 c2 + c21 + c22 + 2b1 d1 + 2b2 d2 + d21 + d22 − 2a2 c1 + 2a1 c2 + c21 + c22 − 2b2 d1 + 2b1 d2 + d21 + d22 ≥ 0, but, as |δ|2 + |γ|2 ≤ |α|2 + |β|2 , the left hand side of 5.4 is less than or equal to a21 + a22 + b21 + b22 + 2a1 c1 + 2a2 c2 + 2b1 d1 + 2b2 d2 − 2a2 c1 + 2a1 c2 + c21 + c22 − 2b2 d1 + 2b1 d2 + d21 + d22 , i.e., a21 + a22 + b21 + b22 + 2a1 c1 + 2a2 c2 + 2b1 d1 + 2b2 d2 − 2a2 c1 + 2a1 c2 + c21 + c22 − 2b2 d1 + 2b1 d2 + d21 + d22 ≥ 0.
68
(5.4)
If we have “strictly greater than” in the above, we get a contradiction with (5.1). Equality requires that |δ|2 + |γ|2 = |α|2 + |β|2 . Since |δ|2 ≤ |α|2 and |γ|2 ≤ |β|2 , equality can only happen if |δ|2 = |α|2 and |γ|2 = |β|2 , which contradicts the fact that αγ − βδ = 1. Thus we must have either |α + δ|2 + |β + γ|2 or |iα + δ|2 + |iβ + γ|2 is less than |α|2 + |β|2 . The other cases are treated similarly. For the first step, i.e. |k|2 = 2, we can apply Claim 5.2.2 above as follows: Let t1 ∈ {k1 , k2 i} be such that |δ + t1 α|2 + |γ + t1 β| < |α|2 + |β|2 and let t2 be the other element of {k1 , k2 i}. Then Ã
t2 α δ + t1 α t2 β γ + t1 β
!
à T =
t2 α δ + t1 α
!Ã
t2 β γ + t1 β
1 1
!
à =
0 1
t2 α δ + kα
!
t2 β γ + kβ
and Ã
t2 α δ + t1 α t2 β γ + t1 β
!
à T0 =
t2 α δ + t1 α
!Ã
t2 β γ + t1 β
1 0
!
à =
1 1
δ + kα δ + t1 α γ + kβ γ + t1 β
Since t2 = im for some integer m, we can apply I − J to M until we have à M=
Defining
and
à M 00 =
! .
t2 β γ + kβ Ã
M0 =
t2 α δ + kα
t2 α δ + t1 α
!
t2 β γ + t1 β δ + kα δ + t1 α γ + kβ γ + t1 β
69
! ,
! .
we then have à M 0 (I − T − T 0 ) = à =
t2 α δ + t1 α t2 β γ + t1 β t2 α δ + t1 α
! (I − T − T 0 ) !
à −
t2 β γ + t1 β
t2 α δ + kα t2 β γ + kβ
!
à −
δ + kα δ + t1 α
!
γ + kβ γ + t1 β
= M 0 − M − M 00 , so, modulo J , we may replace M by M 0 − M 00 . Note that M 0 can be replaced by à Mk0 =
α δ + k0α
! ,
β γ + k0β
with |k 0 |2 = 1, a case we will treat shortly. Also, we have L(M 00 ) ≤ L(M ) and M 00 (∞) 6= α/β 6= M 00 (0). Recall that when L(W 0 ) = L(W ), we need m(W 0 ) < m(W ). In this step, we are replacing one matrix M with two, M 0 and M 00 , but since |δ + t1 α|2 + |γ + t1 β| < |α|2 + |β|2 , we have m(M ) = m(M 0 − M 00 ). (Note that M 00 (∞) = (δ + kα)/(γ + kβ) was already in L(W 0 ) as M 00 (∞) = M (0).) We are not decreasing m(W ) in this step, but we are not increasing it either. In later steps we will obtain a decrease in either L(W 0 ) or m(W 0 ). So now the only matrices M remaining in the support of W with M (∞) = α/β are those of the form
à Mk =
α δ + kα β γ + kβ
! ,
(5.5)
where either k = 0 or |k|2 = 1 and |δ + kα|2 + |γ + kβ|2 ≤ |α|2 + |β|2 (by assumption for those starting out with |k|2 = 1 and by design for those coming initially from matrices with |k|2 = 2). Now we will use elements of J to eliminate the Mk for |k| = 1. In particular, we will use I − T − T 0 . We can first use repeated applications of I − J to replace Mk by
70
˜ k with left hand column given by the transpose of (−kα −kβ), i.e., M Ã ˜k = M Defining
à 0
M =
−kα δ −kβ γ
−kα δ + kα
!
−kβ γ + kβ
!
à 00
and M =
.
δ δ + kα
!
γ γ + kβ
,
we then have à ˜ k (I − T − T 0 ) = M à =
−kα δ + kα −kβ γ + kβ −kα δ + kα
! (I − T − T 0 ) !
à −
−kβ γ + kβ
−kα δ −kβ γ
!
à −
δ δ + kα
!
γ γ + kβ
˜ k − M 0 − M 00 . =M ˜ k by M 0 + M 00 , where M 0 can be replaced by M0 and M 00 Thus we can replace M will satisfy: 1. M 00 (∞) 6=
α β
6= M 00 (0) and
2. L(M 00 ) ≤ L(W ). ˜ k ) = m(M 0 + M 00 ). (Note that M 00 (0) = (δ + kα)/(γ + kβ) Furthermore we have m(M ˜ k (0).) was already in L(W 0 ) as M 00 (0) = M Finally the only matrix in the support of W 0 with either M (∞) or M (0) equal to α/β is M0 . Thus we must have uM0 = 0 and so m(W 0 ) is decreased by at least one.
5.3
Manin Symbols
Recall that in Section 5.1, we showed that we wish to compute H0 (Γ, St ⊗ M ) = (S0 ⊗ M )Γ /∂((S1 ⊗ M )Γ ), 71
where ∂ : (S1 ⊗ M )Γ −→ (S0 ⊗ M )Γ is the boundary map defined by ∂([v1 , v2 , v3 ]) = [v1 , v3 ] − [v1 , v2 ] − [v2 , v3 ]. If M = R, we are computing the set of formal R-linear sums of symbols [v] = [v1 , v2 ], where each vi is a unimodular column in O2 modulo the R[Γ]-module generated by the following elements: 1. [v2 , v1 ] + [v1 , v2 ]; 2. [v] if det(v) = 0; 3. [v1 , v3 ] − [v1 , v2 ] − [v2 , v3 ]; and 4. [v] − γ[v] for all γ ∈ Γ, where the vi run over all unimodular columns in O2 . We call the [v] = [v1 , v2 ] modular symbols. Note that if we compute the same thing but without the [v] − γ[v] relations, then we are just computing the Steinberg module St. We will follow the approach of Wiese [Wie05, p.25-29] in using Proposition 5.2.1 from Section 5.2 to relate Manin symbols to modular symbols. Manin symbols provide us with an explicit, computationally friendly description of the homology we wish to compute. Proposition 5.3.1 below gives us the first step in the transition from modular symbols to Manin symbols. We define another matrix: Ã L :=
We will also use
1 Ã
L2 =
1 −1
! .
0
0 −1 1 −1
! .
In the following we will use the notation α/β for the unimodular column [α β]T . In particular, we will use 0 to denote [0 1]T and ∞ to denote [1 0]T . 72
Proposition 5.3.1. The following homomorphism of R-modules is an isomorphism: Φ : R[GL2 (OK )]/I −→ St M 7−→ [M (0), M (∞)] where I = R[GL2 (OK )](I − J) + R[GL2 (OK )](I + S) + R[GL2 (OK )](I + L + L2 ). Proof. To prove that Φ is surjective, first note that, in the Steinberg module St, we have [v1 , v2 ] = [v1 , 0] + [0, v2 ] = −[0, v1 ] + [0, v2 ], so it suffices to show [0, v 0 ] is in the image of Φ, where v 0 is any unimodular column in O2 . To see this, we use continued fractions to write down a sum of matrices in SL2 (OK ) which, under Φ, will be mapped to [0, v 0 ]. For this algorithm, we rely on the fact that our fields K are Euclidean. We use continued fractions to write a given modular symbol of the form [0, α] as a finite sum of symbols of the form [γ(0), γ(∞)] with γ ∈ SL2 (OK ). We set r0 = α and rn = 1/(rn−1 − an−1 ) and define an = floor(rn ). Here we define the floor function on Q(i) as follows: floor(r + si) for r, s ∈ Q is equal to a + bi with a and b the least integers such that |a − r| ≤ 1/2 and |b − s| ≤ 1/2. We then define the convergents of the continued fractions as follows: p−2 = 0
q−2 = 1
p−1 = 1
q−1 = 0
pn = an pn−1 + pn−2 so that
qn = an qn−1 + qn−2
1 pn = a0 + qn a1 + a2 + 1
1 ···+ a1 n
73
and α=
pk . qk
As in continued fractions for Z, we have pn qn−1 − pn−1 qn = (−1)n+1 . We can now rewrite the modular symbol [0, α] as ¸ k · k k X X X pn−1 pn [0, α] = , = [γn (0), γn (∞)] = Φ(γn ), q q n−1 n n=−1 n=−1 n=−1 where
à γn =
(−1)n+1 pn pn−1 (−1)n+1 qn qn−1
! .
We now show that the kernel of Φ is equal to I = R[GL2 (OK )](I − J) + R[GL2 (OK )](I + S) + R[GL2 (OK )](I + L + L2 ). We start by showing that ker(Φ) = ker(Ψ) where Ψ is the map in Proposition 5.2.1. Define another map π by π : St −→ R[P1 (K)] [v1 , v2 ] 7−→ v2 − v1 . Then we have Ψ = π ◦ Φ, so certainly ker(Φ) ⊆ ker(Ψ). To show the other inclusion, P suppose M uM M ∈ ker(Ψ), i.e., Ψ
à X M
! uM M
=
X
uM M (0) −
M
X M
74
uM M (∞) = 0.
Applying Φ instead of Ψ to this element of ker(Ψ), we then have Φ
à X
! uM M
=
M
X
uM [M (0), M (∞)]
M
=
X
uM [M (0), ∞] +
M
=
X
X
uM [∞, M (∞)]
M
uM [M (0), ∞] −
M
X
uM [M (∞), ∞]
M
= 0, and so ker(Ψ) ⊆ ker(Φ). Finally, we need to show that ker(Φ) can be written in the form claimed, i.e., that ker(Φ) = R[GL2 (OK )](I − J) + R[GL2 (OK )](I + S) + R[GL2 (OK )](I + L + L2 ). Currently, we have that ker(Φ) = R[GL2 (OK )](I − J) + R[GL2 (OK )](I + S) + R[GL2 (OK )](I − T − T 0 ). First, note that in R[GL2 (OK )](I − J) + R[GL2 (OK )](I + S) we have both I + S˜ and I − J˜ where à S˜ = JS =
0 −1 1
0
!
à and J˜ = J 2 =
as I + S˜ = (I − J) + J(I + S) and I − J˜ = (I − J) + J(I − J).
75
−1 0 0
1
!
We then also have I + J˜S˜J˜ as ˜ + J(I ˜ + S) ˜ − J˜S(I ˜ − J). ˜ I + J˜S˜J˜ = (I − J) Now, to see that the two forms of the kernel are the same, we write ˜ + T 0 (I + S) ˜ = I + T S˜ + T 0 S˜ (I − T − T 0 ) + T (I + S) = I + L + L2 and
˜ = I − LJ˜S˜J˜ − L2 J˜S˜J˜ (I + L + L2 ) − (L + L2 )(I + J˜S˜J) = I − T − T 0.
Our discussion thus far is only sufficient for computing weight two modular forms. We now extend this so that we can compute higher weight forms and we also incorporate the Γ relations [v] − γ[v]. ¯ ` and let M = V be a Serre weight with left R[GL2 (OK )] action, Let R = F as defined in Section 3.1. We consider the module Γ (R[GL2 (OK )] ⊗R M ), where Γ acts diagonally on the left and we have the natural right R[GL2 (OK )] action, i.e., (h ⊗ v)g = (hg ⊗ v). In the following theorem and subsequent proposition we use Proposition 5.3.1 to write the homology group we wish to compute in terms of Manin symbols. Theorem 5.3.2. Let N denote the R-module Γ (R[GL2 (OK )] ⊗R M ) as described above. Then the following sequence of R-modules is exact: 0 → N (I − J) + N (I + S) + N (I + L + L2 ) → N → H0 (Γ, St ⊗ M ) → 0. Proof. Proposition 5.3.1 gives the exact sequence 0 → I → R[GL2 (OK )] → St → 0
76
where I = R[GL2 (OK )](I − J) + R[GL2 (OK )](I + S) + R[GL2 (OK )](I + L + L2 ). Since M is a free R-module, the following sequence of R[Γ]-modules is also exact: 0 → N 0 (I − J) + N 0 (I + S) + N 0 (I + L + L2 ) → N 0 → St ⊗ M → 0. We then need only take Γ-coinvariants to achieve the desired exact sequence. We need one more step to get to the module we will actually be computing. This is the content of the following proposition. Proposition 5.3.3. Let X denote the R-module R[Γ\GL2 (OK )] ⊗R M with right GL2 (OK )-action given by (Γh⊗v)g = Γhg ⊗g −1 v. Then there is a right R[GL2 (OK )]module isomorphism between N =Γ (R[GL2 (OK )] ⊗R M ) and X. Proof. The isomorphism is simply given by g ⊗ v 7→ g ⊗ g −1 v. The R-module X = R[Γ\GL2 (OK )] ⊗R M is the module of Manin symbols and is the basic module we will use in computations. For Γ = Γ0 (n), the coset representatives of Γ\GL2 (OK ) are in one-to-one correspondence with P1 (n), the projective line over OK /n. We use the notation (c : d) with c, d ∈ OK to denote such a coset representative.
5.4
Γ1 (n) and Characters
The treatment in Section 5.3 is sufficient for dealing with Γ = Γ0 (n). We will often want to compute forms for Γ1 (n), both for theoretical and computational reasons. The above method can also be used for the Γ1 (n) case (Figueiredo, in [Fig99], does in fact use this method). However, we will follow the approach used by Wiese [Wie05] for the Γ1 (n) case, as it is computationally advantageous to do so. In this section, we describe this method.
77
In the Γ1 (n) case, we will use a character ∼ ¯ ∗. ε : Γ0 (n) ³ Γ1 (n)\Γ0 (n) → (OK /n)∗ → F `
Remarks 5.4.1. 1. When the modular form in question is associated to a Galois representation ρ, this character will correspond to the character ερ of that representation. 2. We recover the Γ0 (n) case by taking Γ1 (n) with the trivial character. We define a slight variation on the weight module M , which takes into account the action of the character ε. This we define as ¯ ε, M ε = M ⊗F¯` F ` ¯ ε denotes a copy of F ¯ ` with action of Γ0 (n) by ε−1 . In particular, for imaginary where F ` quadratic fields K we have ¯ ε. M ε = detaτ ⊗ detaτ 0 ⊗ Symbτ −1 ⊗ Symbτ 0 −1 ⊗ F ` When ` splits in K, the embeddings τ and τ 0 will be the embeddings of kp and k¯p in ¯ ` . When ` is inert in K, the embedding τ 0 will be τ ◦ Frob` . F Computing cohomological mod ` modular forms for Γ1 (n) with character ε then amounts to computing simultaneous eigenvectors for the Hecke operators on ¡ Γ1 (n)\Γ0 (n) Γ1 (n)
¢ (R[GL2 (OK )] ⊗ M ε ) ,
modulo the relations used in Proposition 5.3.1. Here Γ1 (n)\Γ0 (n) is acting diagonally ¡ ¢ ¯ε . on the left on Γ (n) R[GL2 (OK )] ⊗ M ⊗ F 1
`
Proposition 5.4.2. Consider the R-module ¯ ε, X = R[Γ0 (n)\GL2 (OK )] ⊗ M ⊗ F ` 78
where GL2 (OK ) acts on the right by (h ⊗ v ⊗ r)g = (hg ⊗ g −1 v ⊗ r) and Γ1 (n)\Γ0 (n) acts on the left by g(h ⊗ v ⊗ r) = (gh ⊗ v ⊗ ε(g)r). To compute cohomological mod ` modular forms for Γ1 (n) with character ε of the form described above, we can equivalently compute simultaneous eigenvectors for the Hecke operators on the space X modulo the relations used in Proposition 5.3.1. Proof. First we apply the isomorphism ¡ Γ1 (n)\Γ0 (n) Γ1 (n)
¢ (R[GL2 (OK )] ⊗ M ε ) ∼ =Γ0 (n) (R[GL2 (OK )] ⊗ M ε ) .
Then we apply the isomorphism of Proposition 5.3.3.
5.5
Hecke Operators
In this section, we define Hecke operators on the space of modular symbols H0 (Γ, St⊗ ¯ ε ), with Γ = Γ1 (n), but also with left coinvariants by Γ1 (n)\Γ0 (n) via the M ⊗F `
character ε, as described in Section 5.4. (To compute Hecke operators, we convert Manin symbols to modular symbols, compute the action of the Hecke operators there, and then convert back to Manin symbols. We use the results of Section 5.3 to convert back and forth.) Let p be a prime ideal of OK which is relatively prime to the level n and let π be a generator for p. We define a set ∆p ⊂ GL2 (K) by (Ã ∆p =
a b c d
!
à ∈ M2 (OK ) : ad − bc = π,
a b c d
!
à ≡
u ∗ 0 π
!
) mod n ,
∗ where u ∈ OK . Using the fact that K is Euclidean, one can easily show that
Γ1 (n)∆p = ∆p Γ1 (n) = ∆p
79
and ∆p can be written as a disjoint union as
∆p =
[
à Γ1 (n) · σa
a,x
à where σa ≡
1/a 0
a
x
0 π/a
! ,
!
mod n, a ∈ {1, π}, and x runs over representatives of 0 a OK /(π/a). Furthermore, since we take coinvariants via the character action on Γ1 (n)\Γ0 (n) and σa ∈ Γ0 (n), we may define the Hecke operator Tp by à Tp ([v1 , v2 ] ⊗ P ⊗ Q) =ε(π)
+ x
π 0
!
([v1 , v2 ] ⊗ P ⊗ Q) 0 1 Ã ! X 1 x ([v1 , v2 ] ⊗ P ⊗ Q) . 0 π mod π
The Hecke operator Tp is well-defined since Γ1 (n)∆p = ∆p Γ1 (n). Also, the Hecke operator Tp is independent of the choice of generator π, since any other generator will ! Ã ² 0 ∗ be of the form ²π for ² ∈ OK ∈ Γ1 (n). and then T²π = JTπ where J = 0 1
80
Computational Evidence 6.1
Algorithm for Computing Modular Forms
To compute these higher weight mod ` modular forms over Q(i), I wrote a program in C using the PARI library [The05]. In this section, I give an outline of how the program works. Let Γ = Γ1 (n) for some ideal n ⊂ OK with character ∼ ¯ ∗. ε : Γ\Γ0 (n) → (OK /n)∗ → F `
Let V be a Serre weight for K = Q(i) and denote by V ε this same Serre weight but with character action, so we have ¯ ε. V ε = detaτ ⊗ detaτ 0 ⊗ Symbτ −1 ⊗ Symbτ 0 −1 ⊗ F ` The program computes cohomological
mod ` modular forms of level n, character ² ¯ ` . The program executes the and weight V . For notational convenience, set R = F following steps. 1. Compile a list of basic Manin symbols. The Manin symbols module, described in Section 5.3 is the R-module X = R[Γ0 (n)\GL2 (OK )] ⊗R V ε . To compute generators for this module, we compute a set of coset representatives (c : d) for Γ0 (n)\GL2 (OK ), which is in one-to-one correspondence with P1 (n), the projective line over OK /n (see, e.g., [Byg98, p 29]). Recall that each Symb can be viewed as the space of homogeneous polynomials of degree b in two variables with coefficients in R. We can take as a set of generators elements of
81
the form (c : d) ⊗ X m Y n ⊗ Z r W s , where m + n = bτ − 1 and r + s = bτ 0 − 1. In practice, we only store the coset representatives (c : d) and then use a “column number” to indicate the weight (ordered by the exponent of Y and then the exponent of W ). 2. Quotient out the above module by the two and three term relations described in Section 5.3, namely I − J, I + S and I + L + L2 . For this we create a 3m × m matrix, where m is the number of basic Manin symbols computed in Step 1. For each basic Manin symbol, we have three rows, one for each relation. We have to be careful with the character action here. For each row, after computing the action of one of the relations on a basic Manin symbol, we have to write the result again in terms of the basic Manin symbols (for the particular coset representatives we chose for R[Γ0 (n)\GL2 (OK )]). Here we use the left Γ1 (n)\Γ0 (n) action described in Proposition 5.4.2. For example, if after computing the action of one of these relations, we have some (c0 : d0 ) ⊗ X m Y n ⊗ Z r W s in the result, with (c0 : d0 ) = h(c : d) for h ∈ Γ1 (n)\Γ0 (n) and (c : d) ⊗ X m Y n ⊗ Z r W s one of our chosen coset representatives, then we write (c0 : d0 ) ⊗ X m Y n ⊗ Z r W s = ε−1 (h)((c : d) ⊗ X m Y n ⊗ Z r W s ). We then row reduce this matrix to find generators for the space H0 (Γ, St ⊗ V ), as well as linear combinations of these generators for each of the basic Manin symbols. This matrix can be quite large and the reduction of it presents the biggest bottleneck for the program in terms of speed and memory. 3. Once we have the generators for the space and the linear combinations for the basic Manin symbols, we are ready to compute Hecke operators. We compute each Hecke operator Tp as a matrix acting on the generators of the space. Our method of computing Hecke operators is as follows. Take each generator, 82
of the form (c : d) ⊗ X m Y n ⊗ Z r W s and convert it to a modular symbol, i.e., an element of the module ¯ ε ). H0 (Γ, St ⊗ M ⊗ F ` Ã We convert it by first lifting the coset representative (c : d) to a matrix
a b
!
c d GL2 (OK ). (Actually, we can take the lift in SL2 (OK ).) We then convert this to a modular symbol by the following map ÃÃ
a b
!
! ⊗ X mY n ⊗ Z r W s
7→ c d µ· ¸ ¶ b a m n r s ¯ ) (−¯ , ⊗ (bX − dY ) (−cX + aY ) ⊗ (¯bZ − dW cZ + a ¯W ) d c Ã
a b
!
has determinant 1, we do not need to c d account for the action via the deta components of the weight module V .
Note that, since the matrix
We then compute the left action of the Hecke operator Tp on this modular symbol, resulting in a sum of modular symbols. Using basic relations of the modular symbols, we write each as a sum of the form: µ·
¸ ¶ b a , ⊗ P (X, Y ) ⊗ Q(Z, W ) d c ¶ ³h a i ´ µ· b ¸ = 0, ⊗ P (X, Y ) ⊗ Q(Z, W ) − 0, ⊗ P (X, Y ) ⊗ Q(Z, W ) c d
We then use the continued fractions method to convert each of these terms to a sum of basic Manin symbols which in turn can be written in terms of the generators. When converting from modular symbols back to Manin symbols, we again have to be careful with the weight action. The result gives one row of the Hecke operator Tp . We repeat this process for 83
∈
each generator to get the full matrix for Tp .
6.2
Torsion
Early in section 5.1 we assume that the torsion in Γ is invertible in the commutative ¯ ` , though we ring R. In our case, Γ is a congruence subgroup of GL2 (OK ) and R = F may think of R as Fq where q is some power of `. This assumption about torsion is not a strong assumption, since we can make sure that, as long as the level n is large enough, the congruence subgroup Γ will be torsion free. In our examples, we have K = Q(i) and Γ = Γ1 (n) for some level n. Now suppose A ∈ GL2 (K) \ K ∗ is a torsion element of prime power, i.e., there is some prime p such that Ap = I. Now consider K[A]. We have K[A] ∼ = K[x]/(x2 − T x + D), where T is the trace of A and D is the determinant of A. Since A 6∈ K ∗ , we may assume x2 − T x + D is irreducible and so K[A] is a field, and a quadratic extension of K. Since A is a pth root of unity, we also have K[A] ∼ = K[ζp ]. Then we have a quadratic extension of K of the form K[ζp ]. Since K = Q(i), this implies that p is 2 or 3. We will not use ` = 2 for any of our examples (since 2 is ramified in K and hence not covered by the BDJ conjecture), but we do have some examples with ` = 3, so consider p = 3. Then in the polynomial above we have T = −1 and D = 1. The matrix A is in the congruence subgroup Γ1 (n), so we have à A=
a b c d
!
à ≡
∗ ∗ 0 1
! mod n.
Since the determinant D = 1, this implies that a ≡ 1 mod n. Then T = −1 implies 2 ≡ −1 mod n, which in turn implies that n | 3OK . None of our examples have n | 3OK , and so all of our examples satisfy the condition on torsion in Γ.
84
6.3
Modular Forms Corresponding to Elliptic Curves
In Section 4.1, we determined which elliptic curves over Q(i) of those in [Cre84] were supersingular and had good reduction at ` = 7. For the representations corresponding to these elliptic curves we know the level (from the conductor of the elliptic curve) and we know that the character will be trivial. We computed the set of predicted weights using the BDJ conjecture. In this section we give tables for each of the levels with, for some small primes p, the values ap (E) associated to the elliptic curve and the systems of eigenvalues ap (f ) which we found at the predicted levels and weights. Recall that the values ap (E) associated to the elliptic curve are given by ap = 1 + N m(p) − Mp , where N m(p) is the norm of p and Mp is the number of points on the curve E modulo p (including the point at infinity). For each of the four levels tested here, we found the corresponding systems of eigenvalues in all of the predicted weights.
85
Table 6.1: Elliptic curve, conductor n = 6 + 6i, considered mod 7
p ap(E) ap(f ) 1 − 2i −2 5 1 + 2i −2 5 2 − 3i −2 5 2 + 3i −2 5 1 − 4i 2 2 1 + 4i 2 2 2 − 5i 6 6 2 + 5i 6 6 1 − 6i 6 6 1 + 6i 6 6 4 − 5i −6 1 4 + 5i −6 1 2 − 7i −2 5 2 + 7i −2 5 5 − 6i −2 5 5 + 6i −2 5
p ap(E) ap(f ) 3 − 8i 10 3 3 + 8i 10 3 5 − 8i −6 1 5 + 8i −6 1 4 − 9i 2 2 4 + 9i 2 2 1 − 10i −18 3 1 + 10i −18 3 3 − 10i −2 5 3 + 10i −2 5 7 − 8i 18 4 7 + 8i 18 4 11 −6 1 4 − 11i −6 1 4 + 11i −6 1
86
Table 6.2: Elliptic curve, conductor n = 9 + 7i, considered mod 7
p ap(E) ap(f ) 1 + 2i 0 0 3 4 4 2 − 3i −4 3 1 − 4i 0 0 1 + 4i −6 1 2 − 5i −6 1 2 + 5i 0 0 1 − 6i 2 2 1 + 6i 2 2 4 − 5i −6 1 4 + 5i 6 6 2 − 7i 6 6 2 + 7i −6 1 5 − 6i −10 4 5 + 6i 8 1
p ap(E) ap(f ) 3 − 8i 2 2 3 + 8i 2 2 5 − 8i 12 5 5 + 8i −6 1 4 − 9i 8 1 4 + 9i 8 1 1 − 10i −6 1 1 + 10i −18 3 3 − 10i 2 2 3 + 10i 2 2 7 − 8i 6 6 7 + 8i 18 4 11 −4 3 4 − 11i 6 6 4 + 11i −6 1
87
Table 6.3: Elliptic curve, conductor n = 15, considered mod 7
p ap(E) ap(f ) 1+i −1 6 2 − 3i −2 5 2 + 3i −2 5 1 − 4i 2 2 1 + 4i 2 2 2 − 5i −2 5 2 + 5i −2 5 1 − 6i −10 4 1 + 6i −10 4 4 − 5i 10 3 4 + 5i 10 3 2 − 7i −10 4 2 + 7i −10 4 5 − 6i −2 5 5 + 6i −2 5
p ap(E) ap(f ) 3 − 8i 10 3 3 + 8i 10 3 5 − 8i −6 1 5 + 8i −6 1 4 − 9i 2 2 4 + 9i 2 2 1 − 10i 6 6 1 + 10i 6 6 3 − 10i 14 0 3 + 10i 14 0 7 − 8i 2 2 7 + 8i 2 2 11 −6 1 4 − 11i −6 1 4 + 11i −6 1
88
Table 6.4: Elliptic curve, conductor n = 19 + 9i, considered mod 7
p ap(E) ap(f ) 1 − 2i 3 3 1 + 2i 0 0 3 −5 2 2 − 3i 5 5 1 + 4i 3 3 2 − 5i −6 1 2 + 5i 0 0 1 − 6i 2 2 1 + 6i −7 0 4 − 5i 3 3 4 + 5i −3 4 2 − 7i −12 2 2 + 7i 3 3 5 − 6i −1 6 5 + 6i −1 6
p ap(E) ap(f ) 3 − 8i −16 5 3 + 8i 2 2 5 − 8i 12 5 5 + 8i −15 6 4 − 9i −10 4 4 + 9i 8 1 1 − 10i −15 6 1 + 10i 0 0 3 − 10i 2 2 3 + 10i 11 4 7 − 8i −3 4 7 + 8i 0 0 11 −4 3 4 − 11i 6 6 4 + 11i 3 3
89
6.4
Modular Forms Corresponding to Representations from Polynomials
6.4.1 Dihedral Group D4 In Section 4.2.2 we computed a
mod 5 representation ρ with level n = 29 and
quadratic character ε29 , which factors through a Galois extension L over K = Q(i) with G = Gal(L/K) ∼ = D4 . In the following table we recall the correspondence between orders of elements of G and the traces of the images of those elements under ρ. This representation ρ is a base change from a representation over Q, so we give the traces for both the case where the prime p of K splits over Q and for the case where it is inert over Q. order
trace (split)
trace (inert)
1
2
2
2
0, 3
2
4
0
3
Note that in the split case there is some ambiguity in the trace for order 2 elements, depending on whether the element is central. For each of those, I computed the restriction of the representation to the decomposition group at that prime to determine whether it should be 0 or 3. In the following table we list, for some small primes p, the order of Frobp as discussed above along with the coefficients ap of the corresponding system of eigenvalues found. For this example, we found the modular form for two of the four predicted weights. Further investigation is required to understand why the form did not show up as predicted in two of the weights. Table 4.3 in Section 4.2.2 lists the predicted weights along with the level, character and ` for each modular form predicted by the conjecture to correspond to ρ.
90
Table 6.5: Galois group D4 , level n = 29, considered mod 5
p o(Frobp) 1+i 4 3 4 2 − 3i 2 2 + 3i 2 1 − 4i 4 1 + 4i 4 1 − 6i 4 1 + 6i 4 4 − 5i 2 4 + 5i 2 7 2 2 − 7i 2 2 + 7i 2 5 − 6i 2 5 + 6i 2 3 − 8i 4 3 + 8i 4
ap 0 3 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0
p o(Frobp) 5 − 8i 2 5 + 8i 2 4 − 9i 4 4 + 9i 4 1 − 10i 2 1 + 10i 2 3 − 10i 1 3 + 10i 1 7 − 8i 4 7 + 8i 4 11 2 4 − 11i 4 4 + 11i 4 7 − 10i 1 7 + 10i 1
91
ap 0 0 0 0 0 0 2 2 0 0 2 0 0 2 2
6.4.2 Alternating Group A4 In Section 4.2.3 we computed two
mod 3 representations, one of level n = 61 and
the other of level n = 79. Both have trivial character and factor through a Galois extension M over K = Q(i) with G = Gal(M/K) ∼ = Aˆ4 . In the following table we recall the correspondence between orders of elements of G and the traces of the images of those elements under ρ. This representation ρ is a base change from a representation over Q, so we give the traces for both the case where the prime p of K splits over Q and for the case where it is inert over Q. order
trace (split)
trace (inert)
1
2
2
2
1
2
3
2
2
4
0
1
6
1
2
In the following tables we list, for some small primes p, the order of Frobp as discussed above along with the coefficients ap of the corresponding system of eigenvalues found. For both examples, we found the modular form for all the predicted weights. Tables 4.4 and 4.5 in Section 4.2.3 list the predicted weights along with the level, character and ` for each modular form predicted by the conjecture to correspond to ρ.
92
Table 6.6: Galois group A4 , level n = 61, considered mod 3
p o(Frobp) 1+i 4 1 − 2i 6 1 + 2i 6 2 − 3i 6 2 + 3i 6 1 − 4i 6 1 + 4i 6 2 − 5i 3 2 + 5i 3 1 − 6i 4 1 + 6i 4 4 − 5i 4 4 + 5i 4 7 3 2 − 7i 4 2 + 7i 4
ap 0 1 1 1 1 1 1 2 2 0 0 0 0 2 0 0
p o(Frobp) 3 − 8i 3 3 + 8i 3 5 − 8i 4 5 + 8i 4 4 − 9i 3 4 + 9i 3 1 − 10i 6 1 + 10i 6 3 − 10i 6 3 + 10i 6 7 − 8i 4 7 + 8i 4 11 4 4 − 11i 6 4 + 11i 6 7 − 10i 4 7 + 10i 4
93
ap 2 2 0 0 2 2 1 1 1 1 0 0 1 1 1 0 0
Table 6.7: Galois group A4 , level n = 79, considered mod 3
p o(Frobp) 1+i 4 1 − 2i 4 1 + 2i 4 2 − 3i 3 2 + 3i 3 1 − 4i 4 1 + 4i 4 2 − 5i 6 2 + 5i 6 1 − 6i 6 1 + 6i 6 4 − 5i 3 4 + 5i 3 7 3 2 − 7i 6 2 + 7i 6 5 − 6i 3 5 + 6i 3
ap 0 0 0 2 2 0 0 1 1 1 1 2 2 2 1 1 2 2
p o(Frobp) 3 − 8i 3 3 + 8i 3 5 − 8i 2 5 + 8i 2 4 − 9i 6 4 + 9i 6 1 − 10i 3 1 + 10i 3 3 − 10i 6 3 + 10i 6 7 − 8i 6 7 + 8i 6 11 3 4 − 11i 6 4 + 11i 6 7 − 10i 4 7 + 10i 4
94
ap 2 2 1 1 1 1 2 2 1 1 1 1 2 1 1 0 0
6.5
Modular Forms Corresponding to Representations from CFT
6.5.1 Dihedral Group D3 For these D3 examples over K = Q(i), we compute the values tr(ρ(Frobp )) for each prime p by computing the product of 1. the inertia degree of p ⊂ OK in the quadratic extension L, and 2. the order of a prime P ⊂ OL above p in the class group of L. This product gives us the order of Frobp in the Galois group, which is isomorphic to ¯ ` ), D3 . We denote this order by o(Frobp ). Recall that, from the image of ρ in GL2 (F we have the following correspondence between orders of elements and the traces of the images of the elements under ρ: order
trace
1
2
3
−1
2
0
In the following tables we list, for some small primes p, the order of Frobp as discussed above along with the coefficients ap of the systems of eigenvalues (considered mod 5 and
mod 7). There are three tables, one for each of the levels n = 8 + 17i,
n = 13 + 28i and n = 8 + 35i. In all cases, we found the corresponding systems of eigenvalues in all weights predicted for both ` = 5 and for ` = 7. Table 4.6 in Section 4.3.1 lists the predicted weights along with the level, character and ` for each modular form predicted by the conjecture to correspond to ρ.
95
Table 6.8: Galois group D3 , level n = 8 + 17i, considered mod 5 and mod 7
p o(Frobp) 1+i 3 1 − 2i 2 1 + 2i 3 3 2 2 − 3i 3 2 + 3i 2 1 − 4i 3 1 + 4i 3 2 − 5i 2 2 + 5i 2 1 − 6i 3 1 + 6i 2 4 − 5i 2 4 + 5i 2 7 2 2 − 7i 2 2 + 7i 1 5 − 6i 2 5 + 6i 2
ap −1 0 −1 0 −1 0 −1 −1 0 0 −1 0 0 0 0 0 2 0 0
p o(Frobp) 3 − 8i 2 3 + 8i 2 5 − 8i 2 5 + 8i 1 4 − 9i 3 4 + 9i 1 1 − 10i 3 1 + 10i 2 3 − 10i 3 3 + 10i 3 7 − 8i 2 7 + 8i 2 11 3 4 − 11i 3 4 + 11i 2 7 − 10i 2 7 + 10i 3
96
ap 0 0 0 2 −1 2 −1 0 −1 −1 0 0 −1 −1 0 0 −1
Table 6.9: Galois group D3 , level n = 13 + 28i, considered mod 5 and mod 7
p o(Frobp) 1+i 3 1 − 2i 2 1 + 2i 3 3 2 2 − 3i 1 2 + 3i 3 1 − 4i 2 1 + 4i 2 2 − 5i 3 2 + 5i 3 1 − 6i 3 1 + 6i 3 4 − 5i 2 4 + 5i 2 7 3 2 − 7i 2 2 + 7i 2 5 − 6i 3 5 + 6i 2
ap −1 0 −1 0 2 −1 0 0 −1 −1 −1 −1 0 0 −1 0 0 −1 0
p o(Frobp) 3 − 8i 2 3 + 8i 2 5 − 8i 2 5 + 8i 3 4 − 9i 3 4 + 9i 2 1 − 10i 1 1 + 10i 2 3 − 10i 2 3 + 10i 2 7 − 8i 2 7 + 8i 2 11 2 4 − 11i 3 4 + 11i 2 7 − 10i 2 7 + 10i 1
97
ap 0 0 0 −1 −1 0 2 0 0 0 0 0 0 −1 0 0 2
Table 6.10: Galois group D3 , level n = 8 + 35i, considered mod 5 and mod 7
p o(Frobp) 1+i 2 1 − 2i 3 1 + 2i 3 3 2 2 − 3i 2 2 + 3i 3 1 − 4i 3 1 + 4i 2 2 − 5i 2 2 + 5i 2 1 − 6i 3 1 + 6i 2 4 − 5i 3 4 + 5i 1 7 3 2 − 7i 3 2 + 7i 1 5 − 6i 2 5 + 6i 3
ap 0 −1 −1 0 0 −1 −1 0 0 0 −1 0 −1 2 −1 −1 2 0 −1
p o(Frobp) 3 − 8i 3 3 + 8i 1 5 − 8i 2 5 + 8i 3 4 − 9i 3 4 + 9i 2 1 − 10i 1 1 + 10i 3 3 − 10i 2 3 + 10i 3 7 − 8i 1 7 + 8i 2 11 2 4 − 11i 2 4 + 11i 2 7 − 10i 2 7 + 10i 3
98
ap −1 2 0 −1 −1 0 2 −1 0 −1 2 0 0 0 0 0 −1
6.5.2 Dihedral Group D5 In Section 4.3.2 we computed a mod 11 representation ρ with level n = 19 + 20i and quadratic character ε19+20i , which factors through a Galois extension L over K = Q(i) with G = Gal(L/K) ∼ = D5 . In the following table we recall the correspondence between orders of elements of G and the traces of the images of those elements under ρ. order
trace
1
2
2
0
5
3, 7
There are actually two representations: wherever one has a 3 or a 7, the other will have the opposite. Both forms, denoted {ap } and {˜ ap } below, were found for the predicted weights for which we looked. In the following table we list, for some small primes p, the order of Frobp as discussed above along with the coefficients ap and a ˜p of the corresponding systems of eigenvalues found. So far we have only checked two of the three predicted weights, but found both forms in each. Table 4.7 in Section 4.3.2 lists the predicted weights along with the level, character and ` for each modular form predicted by the conjecture to correspond to ρ.
99
Table 6.11: Galois group D5 , level n = 19 + 20i, considered mod 11
p o(Frobp) 1+i 5 1 − 2i 5 1 + 2i 5 3 2 2 − 3i 2 2 + 3i 5 1 − 4i 2 1 + 4i 2 2 − 5i 2 2 + 5i 2 1 − 6i 5 1 + 6i 5 4 − 5i 2 4 + 5i 2 7 2 2 − 7i 2 2 + 7i 5 5 − 6i 5 5 + 6i 2
ap 7 7 3 0 0 3 0 0 0 0 7 7 0 0 0 0 3 7 0
a˜p 3 3 7 0 0 7 0 0 0 0 3 3 0 0 0 0 7 3 0
p o(Frobp) 3 − 8i 2 3 + 8i 5 5 − 8i 2 5 + 8i 2 4 − 9i 2 4 + 9i 5 1 − 10i 5 1 + 10i 5 3 − 10i 1 3 + 10i 2 7 − 8i 2 7 + 8i 2 4 − 11i 5 4 + 11i 5 7 − 10i 5 7 + 10i 5
100
ap 0 7 0 0 0 3 7 3 2 0 0 0 7 3 3 3
a˜p 0 3 0 0 0 7 3 7 2 0 0 0 3 7 7 7
6.6
Final Remarks
In summary, the computational evidence presented here supports the surmise that the BDJ conjectural weight recipe for totally real fields will hold in the case of imaginary quadratic fields as well. For the examples of Galois representations computed here, corresponding modular forms were found in almost all of the predicted weights. There were two exceptions. In one example we did not find the form in all the weights because the computations were too large for the program, so we have not yet looked for all of the predicted weights. In the other exception, we found the form in only two of the four predicted weights. It does look like we may have found a twist of the form in the remaining two weights. Further investigation is required. The modular symbols computation method used in my program is justified here only for K = Q(i). I expect it will be straightforward to use the same methodology for all the other Euclidean class number one imaginary quadratic fields. Note, however, that for each field one must justify an algebraic proposition such as the one presented in Section 5.2. We already know the relations we expect to work for each of these fields, namely those relations computed by Cremona, et al. See, for example, [Cre84]. Several students of Cremona (namely Bygott [Byg98], Lingham [Lin05] and Whitley [Whi90]) have extended the modular symbols method (for trivial weights) to imaginary quadratic fields of higher class number. I expect, with some work, that the method presented in this thesis can be joined with their work to compute modular forms with arbitrary weight for imaginary quadratic fields of higher class number.
101
References [ADP02] A. Ash, D. Doud, and D. Pollack, Galois representations with conjectural connections to arithmetic cohomology, Duke Mathematical Journal 112 (2002), 521–579. [AS00] A. Ash and W. Sinnott, An analogue of Serre’s conjecture for Galois representations and Hecke eigenclasses in the mod p cohomology of GL(n, Z), Duke Mathematical Journal 105 (2000), 1–24. [Ash94] A. Ash, Unstable cohomology of SL(n, O), Journal of Algebra 167 (1994), 330–342. [BCP97] W. Bosma, J. Cannon, and C. Playoust, The Magma algebra system. I. The user language, Journal of Symbolic Computation 24 (1997), no. 3-4, 235–265. [BDJ]
K. Buzzard, F. Diamond, and F. Jarvis, On Serre’s conjecture for mod ` Galois representations over totally real fields, preprint.
[BS73] A. Borel and J.-P. Serre, Corners and arithmetic groups, Commentarii Mathematici Helvetici 48 (1973), 436–491. [Byg98] J. Bygott, Modular forms over imaginary quadratic number fields, Ph.D. thesis, Exeter University, 1998. [Cre84] J. Cremona, Hyperbolic tessellations, modular symbols, and elliptic curves over complex quadratic fields, Compositio Mathematica 51 (1984), 275–323. [S¸08]
M. H. S¸eng¨ un, Serre’s conjecture over imaginary quadratic fields, Ph.D. thesis, University of Wisconsin-Madison, 2008. 102
[DDR] L. Demb´el´e, F. Diamond, and D. Roberts, Numerical examples and evidence for Serre’s conjecture over totally real fields, preprint. [DF+ 97] M. Daberkow, C. Fieker, et al., Kant v4, Journal of Symbolic Computation 24 (1997), 267–283. [Dia97] F. Diamond, The refined conjecture of Serre, Elliptic Curves and Fermat’s Last Theorem (J. Coates and S.-T. Yau, eds.), International Press, Hong Kong, 1997, 2nd. ed., pp. 172–186. [Fig95] L. M. Figueiredo, Serre’s conjecture over imaginary quadratic fields, Ph.D. thesis, University of Cambridge, 1995. [Fig99]
, Serre’s conjecture for imaginary quadratic fields, Compositio Mathematica 118 (1999), 103–122.
[Gee] [Gee07]
T. Gee, On the weights of mod p Hilbert modular forms, preprint. , Companion forms over totally real fields II, Duke Mathematical Journal 136 (2007), 275284.
[Her]
F. Herzig, The weight in a Serre-type conjecture for tame n-dimensional Galois representations, preprint.
[Jan96] G. Janusz, Algebraic number fields, 2 ed., Graduate Studies in Mathematics, vol. 7, American Mathematical Society, USA, 1996. [KM]
J. Kl¨ uners and G. Malle, A database for number fields, available from http://www.math.uni-duesseldorf.de/ klueners/minimum/minimum.html.
[KWa] C. Khare and J.-P. Wintenberger, Serre’s modularity conjecture (I), preprint. [KWb]
, Serre’s modularity conjecture (II), preprint.
[Lin05] M. Lingham, Modular forms and elliptic curves over imaginary quadratic fields, Ph.D. thesis, University of Nottingham, 2005.
103
[LS76]
R. Lee and R. H. Szczarba, On the homology and cohomology of congruence subgroups, Inventiones Mathematicae 33 (1976), no. 1, 15–53.
[Mar01] F. Martin, P´eriodes de formes modulaires de poids 1, Ph.D. thesis, Universit´e Paris 7, 2001. [Mil08] J. S. Milne, Algebraic number theory, 2008, Version 3.00, available from http://www.jmilne.org/math/CourseNotes/ANT.pdf. [Sch]
M. Schein, Weights in Serre’s conjecture for Hilbert modular forms: the ramified case, preprint.
[Ser77] J.-P. Serre, Linear representations of finite groups, Graduate Texts in Mathematics, vol. 42, Springer, New York, 1977. ¯ , Sur les repr´esentations modulaires de degr´e 2 de Gal(Q/Q), Duke
[Ser87]
Mathematical Journal 54 (1987), 179–230. [Sil86]
J. Silverman, The arithmetic of elliptic curves, Graduate Texts in Mathematics, vol. 106, Springer-Verlag, New York, 1986.
[Ste]
W.
Stein,
Sage
mathematical
software
system,
available
from
http://modular.math.washington.edu/sage/doc/reference/. [The05] The PARI Group, Bordeaux, PARI/GP, version 2.3.1, 2005, available from http://pari.math.u-bordeaux.fr/. [Whi90] E. Whitley, Modular forms and elliptic curves over imaginary quadratic fields, Ph.D. thesis, Exeter University, 1990. [Wie05] G. Wiese, Modular forms of weight one over finite fields, Ph.D. thesis, Universiteit Leiden, 2005.
104
Rebecca S. Torrey
A thesis submitted for the degree of Doctor of Philosophy at King’s College London, University of London
Department of Mathematics King’s College London Strand London WC2R 2LS United Kingdom
July 2009
For my mom
Declaration This thesis is a presentation of my own original research work. Wherever contributions of others are involved, every effort has been made to indicate this clearly with due reference to the literature.
Signature: ................................................................. Date: .............................. 3
Abstract In this thesis, I investigate a generalization to imaginary quadratic fields of the refined version of Serre’s conjecture. I ask whether a conjecture analogous to that of Buzzard, Diamond and Jarvis will hold over imaginary quadratic fields. I wrote code to compute cohomological mod ` modular forms over Q(i) of arbitrary weight. I adapt methods of Ash et al., which make use of Borel-Serre duality to express the relevant space of modular forms as a homology group, and then use modular symbols methods generalizing work of Cremona and others. Using an approach of Wiese, I prove algebraically that the modular symbols method will work in this more general setting. With data computed by my program, I provide evidence for Serre’s conjecture in this context.
4
Acknowledgements First and foremost, I would like to take this opportunity to thank my supervisor, Fred Diamond. I am grateful to him for his support, for his unending patience and for sharing with me his wealth of knowledge in this beautiful subject. Other mathematicians whose assistance, advice and work have helped make this thesis possible include Avner Ash, Kevin Buzzard, John Cremona, David Dummit, Frazer Jarvis, Mehmet Haluk S¸eng¨ un and Gabor Wiese. I would also like to thank Manuel Breuning, David Burns and Payman Kassaei, and all the other excellent number theorists in London. It has been an honor to be a part of the London number theory community. My thesis is quite computational in nature, and I owe many thanks to the technical support team of the King’s College maths department, particularly to Dan Wade. I would also like to thank Paul Mitchell for providing computing resources for my code. I started my PhD at Brandeis University and I owe a great deal to many people there. In particular, I would like to thank Ruth Charney, Jennifer Johnson-Leung and Susan Parker. Most of all, I would like to thank my friends and family for their patience, love and support. I have made some great friends among my fellow PhD students, particularly Maria Alicia Lopez Osorio, Chunyan Mu, Aftab Pande and Mark Radosevich. Their humour and support are greatly appreciated. I would like to thank my Go teacher, Guo Juan, and all my Go playing friends. I give thanks to the many friends and family who came to visit me in London, providing little snippets of home, as well as all those who sent their love via post. There are also many old friends whose support from afar I greatly appreciate. I would particularly like to thank my parents Bruce 5
and Kathi and my siblings Mark, Kate, David and Emma and also the matriarch of our wonderful family, Grandma Eileen Noble. Finally, I would like to thank Rich Chalmers without whose patience, support and friendship this would not have been possible.
6
Table of contents Declaration
3
Abstract
4
Acknowledgements
5
1 Introduction
10
2 Classical Serre’s Conjecture
13
2.1
Modular Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13
2.2
Galois Representations . . . . . . . . . . . . . . . . . . . . . . . . . .
15
2.2.1
Level of ρ . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
16
2.2.2
Character of ρ . . . . . . . . . . . . . . . . . . . . . . . . . . .
18
2.3 Serre’s Conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18
3 Generalized Serre’s Conjecture
21
3.1 Serre Weights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
21
3.2 Fundamental Characters . . . . . . . . . . . . . . . . . . . . . . . . .
24
3.3 The Weight Conjecture for Totally Real Fields . . . . . . . . . . . . .
26
3.4 Cohomological mod ` Forms Over K . . . . . . . . . . . . . . . . .
32
3.5 Serre’s Conjecture for Imaginary Quadratic Fields . . . . . . . . . . .
33
4 Examples of Galois Representations
36
4.1 Examples from Elliptic Curves . . . . . . . . . . . . . . . . . . . . . .
36
4.2 Examples from Polynomials . . . . . . . . . . . . . . . . . . . . . . .
39
7
4.2.1
Techniques & Background . . . . . . . . . . . . . . . . . . . .
39
4.2.2
Dihedral Group D4 . . . . . . . . . . . . . . . . . . . . . . . .
43
4.2.3
Alternating Group A4 . . . . . . . . . . . . . . . . . . . . . .
46
4.3 Examples from Class Field Theory . . . . . . . . . . . . . . . . . . .
52
4.3.1
Dihedral Group D3 . . . . . . . . . . . . . . . . . . . . . . . .
52
4.3.2
Dihedral Group D5 . . . . . . . . . . . . . . . . . . . . . . . .
56
5 Computing Modular Forms over Q(i)
59
5.1 Borel-Serre Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . .
59
5.2 An Algebraic Proposition . . . . . . . . . . . . . . . . . . . . . . . . .
62
5.3 Manin Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
71
5.4 Γ1 (n) and Characters . . . . . . . . . . . . . . . . . . . . . . . . . . .
77
5.5
79
Hecke Operators
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6 Computational Evidence
81
6.1
Algorithm for Computing Modular Forms
. . . . . . . . . . . . . . .
81
6.2
Torsion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
84
6.3
Modular Forms Corresponding to Elliptic Curves . . . . . . . . . . .
85
6.4
Modular Forms Corresponding to Representations from Polynomials .
90
6.4.1
Dihedral Group D4 . . . . . . . . . . . . . . . . . . . . . . . .
90
6.4.2
Alternating Group A4 . . . . . . . . . . . . . . . . . . . . . .
92
6.5 Modular Forms Corresponding to Representations from CFT . . . . .
95
6.5.1
Dihedral Group D3 . . . . . . . . . . . . . . . . . . . . . . . .
95
6.5.2
Dihedral Group D5 . . . . . . . . . . . . . . . . . . . . . . . .
99
6.6 Final Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 References
102
8
List of Tables 4.1
Weights for elliptic curves mod 7 . . . . . . . . . . . . . . . . . . . .
37
4.2
Elliptic curves over Q(i) mod 7 with several levels . . . . . . . . . .
39
4.3
D4 representation mod 5 with level n = 29 . . . . . . . . . . . . . .
45
4.4
A4 representation mod 3 with level n = 61 . . . . . . . . . . . . . .
48
4.5
A4 representation mod 3 with level n = 79 . . . . . . . . . . . . . .
51
4.6
D3 representations mod 5 and mod 7 with several levels . . . . . .
55
4.7
D5 representation mod 11 with level n = 19 + 20i . . . . . . . . . .
58
6.1 Elliptic curve, conductor n = 6 + 6i, considered mod 7 . . . . . . . .
86
6.2 Elliptic curve, conductor n = 9 + 7i, considered mod 7 . . . . . . . .
87
6.3 Elliptic curve, conductor n = 15, considered mod 7 . . . . . . . . . .
88
6.4 Elliptic curve, conductor n = 19 + 9i, considered mod 7 . . . . . . .
89
6.5
Galois group D4 , level n = 29, considered mod 5 . . . . . . . . . . .
91
6.6 Galois group A4 , level n = 61, considered mod 3 . . . . . . . . . . .
93
6.7
Galois group A4 , level n = 79, considered mod 3 . . . . . . . . . . .
94
6.8
Galois group D3 , level n = 8 + 17i, considered mod 5 and mod 7 .
96
6.9
Galois group D3 , level n = 13 + 28i, considered mod 5 and mod 7
97
6.10 Galois group D3 , level n = 8 + 35i, considered mod 5 and mod 7 .
98
6.11 Galois group D5 , level n = 19 + 20i, considered mod 11 . . . . . . . 100
9
Introduction The Langlands Program, started in the late 1960s by Robert Langlands, is a system of powerful conjectures connecting number theory to the representation theory of certain groups. It incorporates an earlier construction of Shimura which associates Galois representations to modular forms. Serre’s conjecture, first formulated by Jean-Pierre Serre in a 1987 article [Ser87], provides a converse in characteristic ` to Shimura’s construction. Serre’s conjecture states that any odd, irreducible representation ρ : Gal(K/Q) → GL2 (Fq ) where K is a Galois number field and Fq is a finite field of characteristic `, is modular, i.e., arises from a modular form. A refinement to Serre’s conjecture gives the minimal weight and level of this modular form and, by work of Ribet and others, we know that Serre’s conjecture holds if and only if the refined version holds. Recently, Khare and Wintenberger have completed the proof of Serre’s conjecture using ideas and results of Dieulefait, Kisin, Taylor and Wiles. It is natural to ask whether an analogous conjecture holds for representations of Gal(K/F ) where F is an arbitrary number field. Buzzard, Diamond and Jarvis [BDJ] recently formulated a version of the refined Serre’s conjecture in the case in which F is totally real and ` is unramified, where predicting the weights is much more complicated than for F = Q. In this more general setting, it no longer makes sense to simply specify a minimal weight and level for the corresponding modular form. A more general notion of weight is needed. Also, the refined part of the conjecture takes the form of a recipe for all the weight combinations for modular forms giving rise to a particular representation. This recipe depends only on the local behavior of the representation ρ at primes above `. There has been some progress, due to Gee, towards proving the equivalence of Serre’s conjecture and the refined Serre’s conjecture for totally real fields F , but crucial steps in the method of Khare 10
and Wintenberger break down in this situation. If F is not totally real, then the situation is even less well understood. In this case, we do not even have a complete understanding of how to associate Galois representations to modular forms. This situation is the focus of my research. Some computational evidence is available for generalizations of Serre’s conjecture to number fields. Demb´el´e [DDR] has done computations of arbitrary weight mod ` modular forms over totally real fields F , providing evidence for the conjecture of Buzzard, Diamond and Jarvis. For imaginary quadratic fields, Figueiredo [Fig99] provided some computational evidence for Serre’s conjecture, but he worked only with weight two modular forms. More recently, S¸eng¨ un [S¸08] proved non-existence of certain representations and has done some computations of arbitrary weight modular forms over imaginary quadratic fields. In this thesis, I ask whether a conjecture analogous to that of Buzzard, Diamond and Jarvis will hold over imaginary quadratic fields. I wrote code to compute cohomological mod ` modular forms over Q(i) of arbitrary weight. My approach differs from that of S¸eng¨ un, who computes cohomology. Instead, I compute homology using modular symbols methods generalizing work of Cremona [Cre84] and others. This homological method seems to be more promising for adapting methods developed by Cremona and others for computing forms over imaginary quadratic fields of class number greater than one. In Chapter 2, I review the classical case, i.e. Serre’s conjecture over Q. I give basic definitions and some basic facts about modular forms and Galois representations in this setting. I state Serre’s original conjecture and discuss work that has been done on the conjecture itself and towards generalizations of the conjecture. In Chapter 3, I define Serre weights and review the conjecture of Buzzard, Diamond and Jarvis (BDJ) over totally real fields. I then define mod ` modular forms cohomologically for imaginary quadratic fields and ask whether the BDJ conjecture will hold in the imaginary quadratic case. I compute examples of Galois representations in Chapter 4. Some of these examples arise from polynomials, some are from elliptic curves and others are constructed using class field theory. For each I find the level and character and then compute the 11
predicted weights using the BDJ conjecture. I compute the predicted weights for two of the three examples given in Figueiredo’s thesis. In Chapter 5, I prove that the modular symbols method can be used to compute the space of modular forms in which I am interested. To account for the higher weights, one needs to compute cohomology with non-trivial coefficients. In this case, Cremona’s geometric approach becomes intractable. I adapt methods of Ash et al., which make use of Borel-Serre duality to express the relevant space of modular forms as a homology group. From Borel-Serre [BS73], we have an isomorphism ∼
H 2 (Γ, V ) −→ H0 (Γ, St ⊗ V ), where St denotes the Steinberg module, and V denotes a “Serre weight”, i.e., an ¯ ` -representation of irreducible F G = GL2 (OK /`OK ). Following Ash in [Ash94], I describe the Steinberg module in terms of universal minimal modular symbols. This allows one to sidestep the geometric argument, and prove algebraically that the modular symbols method can be used to compute H0 (Γ, St⊗V ). In order to obtain a description of the space which can actually be used for computations, one must be able to go from modular symbols to Manin symbols (M-symbols). Without recourse to Cremona’s geometric method, I needed a different way to justify this conversion. Following an approach of Wiese [Wie05], I prove algebraically that one can use M-symbols to compute this homology space for K = Q(i). In Chapter 6, I present computational evidence in support of the conjecture. I summarize the algorithm used in my program to compute the cohomological mod ` modular forms over Q(i). Finally, I give tables of systems of eigenvalues matching the traces of ρ(Frobp ) of the Galois representations ρ computed in Chapter 4.
12
Classical Serre’s Conjecture Serre’s conjecture relates Galois representations to modular forms. In this chapter, we set the stage by reviewing the classical case. We will give basic definitions of modular forms and Galois representations and then state Serre’s original and refined modularity conjectures. Finally we give a brief overview of the current status of Serre’s conjecture (which is now a theorem) and indicate various work undertaken towards understanding the more general situation.
2.1
Modular Forms
We start by giving the classical definition of modular forms (i.e., over Q). Modular forms are functions on the upper half of the complex plane which satisfy certain symmetry and “boundedness” conditions. To describe these conditions in more detail, we first need some notation. Let h = {z ∈ C | Im(z) > 0} be the complex upper half plane. The modular group (Ã SL2 (Z) =
a b c d
!
) | a, b, c, d ∈ Z and ad − bc = 1 Ã
acts on h by fractional linear transformations: for g = z ∈ h, we have g(z) =
az + b . cz + d
13
a b c d
! ∈ SL2 (Z), and
Any subgroup Γ of SL2 (Z) which contains the subgroup (Ã Γ(N ) =
a b
!
à ∈ SL2 (Z) |
c d
a b
!
à ≡
c d
1 0
!
) mod N
0 1
for some positive integer N is called a congruence subgroup. The level of a congruence subgroup Γ is the smallest N such that Γ(N ) ⊆ Γ. We define two important congruence subgroups in particular: (Ã Γ0 (N ) =
a b
!
à ∈ SL2 (Z) |
c d
a b
!
à ≡
c d
∗ ∗
!
) mod N
0 ∗
and (Ã Γ1 (N ) =
a b c d
!
à ∈ SL2 (Z) |
a b
!
c d
à ≡
∗ ∗ 0 1
!
) mod N
.
Definition 2.1.1. A modular form of weight k and level N is a function on h which is holomorphic everywhere (including at the cusps) and which satisfies f (z) = (cz + d)−k f (g(z)), Ã for all g =
a b c d
! in Γ, where Γ is a congruence subgroup of level N .
The modular forms of weight k for a congruence subgroup Γ form a finite dimensional complex vector space which we denote by Mk (Γ). Ã ! 1 1 For Γ = Γ0 (N ) or Γ = Γ1 (N ), we have ∈ Γ, so that a modular form f 0 1 for such a congruence subgroup satisfies f (z) = f (z + 1). This, together with the holomorphicity condition, implies that f has a Fourier ex-
14
pansion of the form f (z) =
∞ X
an q n ,
q = e2πiz .
n=0
à (For any congruence subgroup Γ, we at least have
1 N
!
∈ Γ, giving a similar 0 1 expansion but with q = e2πiz/N .) If a0 = 0 in the Fourier expansion of (cz+d)−k f (g(z)) for all g ∈ SL2 (Z), we call f a cusp form. We denote the space of cusp forms by Sk (Γ). For Γ = Γ1 (N ), one can define certain operators Tn for n ≥ 1, called Hecke operators, and hdi for d ∈ (Z/N Z)∗ , called diamond bracket operators, which act on the space Mk (Γ1 (N )) and on the space Sk (Γ1 (N )). Together these operators form a commutative algebra called the Hecke algebra H = Z[Tn , hdi]. A non-zero cusp form f ∈ Sk (Γ1 (N )) is called an eigenform if it is a simultaneous eigenvector for all operators in the Hecke algebra. If f ∈ Sk (Γ1 (N )) is an eigenform, we define the character of f to be the Dirichlet character ε : (Z/N Z)∗ → C∗ such that f |hdi = ε(d)f . We denote the space of forms with a given character ε by Sk (Γ1 (N ), ε). We define a system of eigenvalues in Sk (Γ1 (N ), ε) to be a set of eigenvalues (an ) of the Hecke operators Tn for n ≥ 1 acting on an eigenform f ∈ Sk (Γ1 (N ), ε). One can show that the following relationships hold for Hecke operators: 1. Tmn = Tm Tn
if gcd(m, n) = 1, and
2. Tpn = Tpn−1 Tp − pk−1 ε(p)Tpn−2
for p prime.
Thus to determine a given system of eigenvalues one need only compute the eigenvalues ap for p prime.
2.2
Galois Representations
The Galois representations of Serre’s Conjecture (and, consequently, those that will be of concern to us) are the mod ` Galois representations. Recall the definition:
15
Definition 2.2.1. A mod ` Galois representation is a continuous homomorphism ¯ ` ). ρ : GQ −→ GLn (F In this thesis, the representations that we will be interested in will be continuous homomorphisms ¯ `) ρ : GK −→ GL2 (F ¯ where K is an imaginary quadratic field and GK = Gal(Q/K). Since we are interested in the refined version of Serre’s conjecture, we will also need the definitions of level, weight and character. In the following two subsections we define the level and character associated to ρ. The definition of the weight kρ is more complicated. Instead of defining it specifically for the classical case, we will discuss the BDJ generalization of the weight recipe in Chapter 3.
2.2.1 Level of ρ To define the level of ρ, we follow Serre’s exposition in his seminal 1987 paper on the subject ([Ser87]). The level is defined to be the Artin conductor of ρ, defined as it is in characteristic 0, except that in this case we take the prime to ` part of the conductor. We will now give the precise definition. First write the Galois representation as ρ : GQ −→ GL(V ), ¯ ` . Let p 6= ` be a rational prime. The where V is a 2-dimensional vector space over F representation ρ will factor through some finite extension L over Q, so we may write ρ : Gal(L/Q) −→ GL(V ). Let G be the Galois group Gal(L/Q). Let D0 ⊃ D1 ⊃ · · · ⊃ Di ⊃ · · · 16
be the higher ramification groups of G corresponding to the prime p. For each i, denote by Vi the subspace of V fixed by Di . Define the integer n(p, ρ) by n(p, ρ) =
∞ X i=0
1 dim(V /Vi ). (D0 : Di )
We have the following properties regarding n(p, ρ). 1. n(p, ρ) = 0 if and only if D0 is trivial, which is if and only if ρ is unramified at p; 2. n(p, ρ) ≥ 1 if ρ is ramified at p; 3. n(p, ρ) = dim(V /V0 ) if and only if D1 is trivial, which is if and only if ρ is tamely ramified at p. We define the level N (ρ) by N (ρ) =
Y
pn(p,ρ) .
p6=`
This thesis is concerned with representations over an imaginary quadratic field K, i.e. ρK : GK → GL(V ). Now ρK will factor through some Galois extension L over K, so the Galois group in question will be G = Gal(L/K). From there on, we can define the level the same way, except that we will take primes p ∈ OK , so that the level n(ρK ) =
Y
pn(p,ρ)
p-`·OK
will be an ideal of OK . The level will still, by definition, be prime to `.
17
2.2.2 Character of ρ In this section, we will define a generalization to imaginary quadratic fields K of the character ερ that Serre associated to ρ in [Ser87]. This character ερ will be obtained from the character ¯ ¯ `) det ρ : GK = Gal(Q/K) → GL2 (F by removing the ` part. We will now make this more precise. Since the conductor of det ρ must divide `n(ρ) (see, e.g., [Fig95, p 7]), we can identify det ρ with a homomorphism ¯ ∗, det ρ : (OK /`n(ρ))∗ → F ` or, equivalently, to a pair of homomorphisms ¯∗ ϕρ : (OK /`OK )∗ → F ` and ¯ ∗. ερ : (OK /n(ρ))∗ → F ` We define the character of ρ to be the character ερ .
2.3
Serre’s Conjecture
In a 1987 paper [Ser87], Serre gave a conjecture prescribing a relationship between modular forms and mod ` Galois representations. Conjecture 2.3.1. (Serre, 1987) Suppose ¯ ` ), ρ : GQ → GL2 (F is a
mod ` Galois representation which is continuous, odd and irreducible. Then P there is a modular form f = an q n such that tr(ρ(Frobp )) = ap and det(ρ(Frobp )) = ε(p)pk−1 for all p - N `. Here N is the level, k the weight and ε the character of f . 18
Whenever such an f does exist for a given ρ, we say that ρ is modular. Furthermore, Serre gave a refinement of his conjecture, in which he prescribed the minimum weight and level of such an eigenform f , assuming k ≥ 2 and ` - N , using the definitions given in the preceding section (though we have omitted the definition of the weight kρ ). Conjecture 2.3.2. (Serre, refined) Suppose ¯ `) ρ : GQ → GL2 (F is a
mod ` Galois representation which is continuous, odd and irreducible. Then ρ
is modular and one can take the corresponding modular form f to be in Skρ (Γ, ερ ) with Γ of level N (ρ). Furthermore, the level N (ρ) and the weight kρ are minimal among levels prime to ` and weights k ≥ 2. Serre’s conjecture is now a theorem. First, work by Ribet, Gross, Coleman-Voloch and others showed that the original conjecture and the refined conjecture are equivalent (see, e.g., [Dia97]). Recently Khare and Wintenberger ([KWa], [KWb]) proved the conjecture itself using work of Dieulefait, Taylor, Wiles and Kisin. This thesis is concerned with a generalization of Serre’s conjecture. There are two natural avenues to explore. One is to consider representations of higher dimension, i.e. representations ¯ ` ), ρ : GQ → GLn (F for n > 2. Progress has been made in developing conjectures and providing evidence for these conjectures by Ash, Doud, Pollack and Sinnott (see [ADP02] and [AS00]) and also by Herzig ([Her]). In Chapter 4 we will look at some examples of 2-dimensional representations considered in the papers by Ash et al. They use these examples to construct reducible 3-dimensional representations, but we will consider irreducible 2-dimensional representations they compute in the process. A second natural generalization of Serre’s conjecture is to consider more general
19
number fields, i.e. representations ¯ ) → GL2 (F ¯ ` ), ρ : GF = Gal(Q/F for an extension F of Q. This type of generalization is the focus of this thesis. In a forthcoming paper, Buzzard, Diamond and Jarvis (BDJ) [BDJ] consider the case of totally real fields F . They formulate a version of Serre’s refined conjecture in this context, which we will review in Section 3.3. Totally real fields are a natural starting place as there is already a complete map in the other direction, i.e. one can associate a Galois representation to a Hilbert modular cusp form. In this situation, it no longer makes sense to talk about minimal weights. Instead ¯ ` representations Buzzard, Diamond and Jarvis define Serre weights to be irreducible F σ of GL2 (O/`O) and explain what it means for a representation to be modular of weight σ. Their conjecture then, instead of giving a minimal weight, describes all the possible Serre weights and level structures of modular forms giving rise to a given representation ρ. They show that their conjectural weight recipe is correct for K = Q. They also cite both theoretical and computational evidence supporting the conjecture. This includes calculations by Demb´el´e, Diamond and Roberts [DDR] of Hilbert modular forms with these weights and level structures and their corresponding Galois representations. The BDJ conjecture covers mod ` representations for unramified primes `. Schein [Sch] has subsequently extended this conjecture to ramified primes ` > 2. In [Gee07] and [Gee], Gee made some progress towards proving the regular and refined versions of Serre’s conjecture are equivalent in the totally real case. In a 1998 paper [Fig99], Figueiredo investigated the possibility of generalizing Serre’s Conjecture to imaginary quadratic fields. He computed
mod ` cusp forms
and corresponding Galois representations, providing evidence that an analogous correspondence would also hold in this situation. His computations are limited to weight 2 and he did not formulate a refined version of Serre’s Conjecture. In this thesis we explore the refined version of Serre’s conjecture over imaginary quadratic fields. 20
Generalized Serre’s Conjecture In this chapter, we discuss the generalization of Serre’s conjecture to number fields. We define Serre weights, a generalization of the classical notion of weights. We review some basics about fundamental characters, which will be used in subsequent sections. We discuss the BDJ conjecture for totally real fields. Finally we define cohomological mod ` forms for imaginary quadratic fields and ask whether the BDJ conjecture will hold in this case as well.
3.1
Serre Weights
Let K be a number field and denote by O its ring of integers. Fix a prime ` which is unramified in K. ¯ ` -representation V of Definition 3.1.1. Define a Serre weight to be an irreducible F G = GL2 (O/`O). In this section we describe what these Serre weights look like and describe the form in which we work with them computationally. Denote by SK the set of embeddings τ : K ,→ C. Set G = GL2 (O/`O) ∼ =
Y
GL2 (O/p).
p|`
For each prime p of K such that p|`, set kp = O/p and fp = [kp : F` ]. We may identify Q ¯ ` . We will see that the SK with p|` Sp , where Sp is the set of embeddings kp ,→ F ¯ ` -representations of G are of the form irreducible F V =
O
Vp
¯ `) (the tensor product is taken over F
p|`
21
where Vp =
O¡
¢ ¯ `. detaτ ⊗kp Symbτ −1 kp2 ⊗τ F
τ ∈Sp
¯ ` -representation V is a tensor product over primes p dividing `; each Note that the F factor Vp will act on the corresponding factor GL2 (O/p) of GL2 (O/`O). For each of these factors Vp , we take the tensor product over all τ ∈ Sp , so we will take a closer look at these embeddings τ . Fix some τ0 ∈ Sp : ¯ `. τ0 : kp = O/p ,→ F Then for each i such that 1 ≤ i < f where f = [kp : F` ], set τi = τ0 ◦ Frobi` . We then have Sp = {τi : 0 ≤ i < f }. For computational purposes, we will think of Symbτ −1 (kp2 ) as the space of homogeneous polynomials of degree bτ − 1 in two variables with coefficients in kp . We define the left action of GL2 (OK ) on this space as follows: for g ∈ GL2 (OK ), we reduce g modulo p and then the action is given by g¯ · P (X, Y ) = P (dX − bY, −cX + aY ), Ã for g¯ =
a b
c d the inverse, i.e.,
! ∈ GL2 (OK /p). We will also need a right action, for which we take
P (X, Y ) · g¯ = (¯ g )−1 · P (X, Y ). Since the Serre weights are irreducible representations, we may assume 1 ≤ bτ ≤ `. If bτ > `, the ideal I generated by X ` and Y ` is stable under the action of G since g(X ` ) ∈ I and g(Y ` ) ∈ I so g(I) ⊆ I. In this case, we get a G-stable subspace of ¯ 2 ), so this is a reducible representation of G. Symbτ −1 (F `
The determinant map is simply multiplication by the determinant (or some power of the determinant) of g for g ∈ G. This is sometimes denoted detaτ kp2 . We can view 22
the representation
O
¯ `) (detaτ ⊗τ F
τ ∈Sp
¯ ` via the embedding (where the inner tensor product is taken over the image of kp in F by that particular τ ) as the tensor product of characters ¯× (τ ◦ det)aτ : GL2 (kp ) −→ F ` for integers aτ . That is, for each embedding τ , we have the diagram ¯× GL2 (kp ) −→ F ` ∪ .
det ↓ τ
−→ F× `f
kp× So now we can write
O
¯ `) = (detaτ ⊗τ F
τ
O
(τ ◦ det)aτ .
τ
Furthermore, writing the various τ ∈ Sp as τi (as defined above), we can express this as
O
(τ ◦ det)
aτ
=
τ
f −1 O
(τi ◦ det)ai ,
i=0
¯ ` . Then where ai = aτi and both tensor products are taken over F f −1 O
f −1 ai
(τi ◦ det) =
i=0
Y
i
(τ0 ◦ det)ai ` = (τ0 ◦ det)
Pf −1 i=0
a i `i
.
i=0
Summarizing, we get that O
¯ ` ) = (τ0 ◦ det) (detaτ ⊗τ F
Pf −1 i=0
a i `i
.
τ
Thus the representation
N
aτ ¯ ` ) is determined by the exponents Pf −1 ai `i (det ⊗ F τ i=0 τ
mod (`f − 1), so we can now see that to get all of these representations, it suffices to allow the ai to run from 0 to ` − 1. Taking all the fp -tuples of ai where 0 ≤ ai ≤ ` − 1, 23
we get all the distinct representations of this type, with one redundancy: if all the ai are 0 we get the same representation as when all the ai are ` − 1. Thus we make the additional assumption that in each fp -tuple of ai , one of the ai is strictly less than ` − 1.
3.2
Fundamental Characters
In this section we review the concept of fundamental characters, as we will need this in the subsequent section where we define the conjectural weight recipe. Let K be a local field of characterstic 0, so K is a finite extension of Qp for some rational prime p. Let L be a finite Galois extension of K and let l and k be the residue fields of L and K respectively. There is a natural surjection of Galois groups Gal(L/K) → Gal(l/k). We define the inertia group of L over K, denoted I(L/K), to be the kernel of this map. For an extension L0 over L we have a surjection I(L0 /K) → I(L/K) and so we can define the inertia group IK to be the inverse limit of the I(L/K) as L ranges over ¯ all finite Galois extensions of K in K. From field theory, we know that for each n there is an extension of k of degree n
n, which we will denote by kn . This extension is the splitting field of X q − X and it is unique up to k-isomorphism. The Galois group of this extension, Gal(kn /k), is cyclic of order n, with canonical generator the Frobenius element x 7→ xq . Milne’s Proposition 7.50 [Mil08, p.121] says the following: Proposition 3.2.1. Let L be an algebraic extension of K. There is a one-to-one inclusion reversing correspondence of sets {K 0 ⊂ L, finite and unramified over K} ↔ {k 0 ⊂ l, finite over k},
24
where k 0 is the residue field of K 0 . Furthermore, there is a canonical isomorphism Gal(K 0 /K) → Gal(k 0 /k). From this proposition and the preceding discussion, we see that for each n there is n
a degree n unramified extension Kn of K. It is again the splitting field of X q −X and it is unique up to K-isomorphism. Since the Galois groups must be isomorphic, we see that Gal(Kn /K) is cyclic of order n. As before the Galois group has as canonical generator the Frobenius element σ which is the element of the Galois group satisfying the property σβ ≡ β q
mod p for all β ∈ OKn .
In subsequent sections, we will need something from local class field theory called ¯ ,→ K ¯ p . This gives us a corresponding a fundamental character. Fix an embedding K inclusion of Galois groups ¯ p /Kp ) ,→ Gal(K/K), ¯ Gal(K
σ 7→ σ|K¯ .
Via this inclusion, we may regard the inertia subgroup IKp as a subgroup of GK = ¯ Gal(K/K). Letting f denote the degree of the extension Kp over Q` , we have that 1
the residue field kp of Kp is isomorphic to F`f . Let Kp0 = Kp ((−`) `f −1 ). The field Kp contains all the (`f − 1)st roots of unity, so the extension Kp0 is Galois over Kp . We also know, from Kummer theory, that Gal(Kp0 /Kp ) = kp× = F× . `f Since Kp is unramified over Q` , the inertia group IKp injects into the Galois group ¯ ` /Kp ) which, in turn, injects into Gal(Q ¯ ` /Q` ). We thus get a map Gal(Q ¯ ` /Kp ) → Gal(K 0 /Kp ) → F×f , IKp ,→ Gal(Q p ` ¯ ` /Kp ) → Gal(K 0 /Kp ) is simply restriction to K 0 . The map where the map Gal(Q p p given above is called the fundamental character. IKp → F× `f 25
3.3
The Weight Conjecture for Totally Real Fields
In this section, we will describe, at least in part, the conjectural weight recipe given by Buzzard, Diamond and Jarvis in [BDJ] for the generalization to totally real fields of the refined version of Serre’s conjecture. Let K be a totally real field and fix a prime ` which is unramified in K. Let ¯ ` ) be a continuous, irreducible and totally odd representation. ρ : GK → GL2 (F As described in Section 3.1, each Serre weight V is of the form V = ⊗F¯` Vp where each Vp is an irreducible representation of GL2 (kp ) and the p run over all the primes of K which divide `. We will describe a set Wp (ρ) of GL2 (kp )-representations; the conjectural weight set W (ρ) will then consist of Serre weights of the form V = ⊗F¯` Vp where each Vp ∈ Wp (ρ). First, we will need some more notation. For the moment, we will be working with a fixed prime p dividing `, so we will suppress the p from the notation, i.e., instead of kp , fp and Sp , we will simply write k, f and S, respectively. We fix an ¯ ,→ K ¯ p and identify GKp and IKp with subgroups of GK . Denote by embedding K ¯ p , and extend this prime notation K 0 the unramified quadratic extension of Kp in K p
to its associated entities, i.e., we have k 0 , f 0 and S 0 . (We only need to consider the quadratic extension because we are only looking at 2-dimensional representations.) Write D = GKp and D0 = GKp0 . Define a projection map π : S 0 → S by τ 0 7→ τ 0 |k . For an embedding σ ∈ S or σ ∈ S 0 , let ωσ denote the fundamental character of IL defined by composing σ with the homomorphism from local class field theory IL → (O/`O)× (here L is either Kp or Kp0 depending on whether σ is in S or S 0 ). The sets Wp (ρ) will only depend on the local behaviour of ρ at ` and we will split the definition into two cases: when ρ|D is reducible and when it is irreducible. First we consider the case where ρ|D is irreducible. For a subset J of S 0 such that ∼
π : J → S, we define a matrix à Q M~b,J =
b
0
π(τ ) τ 0 ∈J ωτ 0
Q
!
0 b
0
π(τ ) τ 0 6∈J ωτ 0
0
26
.
Then we define Wp (ρ) by ( Wp (ρ) =
V~a,~b : ρ|I ∼
Y
) ∼
ωτaτ M~b,J for some J ⊂ S 0 such that π : J → S
.
τ ∈S
In this case, when ρ|D is irreducible, we can write ρ|D as the induction of a character, ¯ × . We use this to define a set W 0 (ξ) i.e., ρ|D ∼ IndD0 ξ for some character ξ : D0 → F D
`
which is very close to Wp (ρ): ( W 0 (ξ) =
∼
(V~a,~b , J) : J ⊂ S 0 , π : J → S, ξ|I =
Y
ωτaτ
τ ∈S
Y
) b 0) ωτπ(τ 0
.
τ 0 ∈J
We have Wp (ρ) = {V |(V, J) ∈ W 0 (ξ) for some J} and so we have a projection map ∼
W 0 (ξ) → Wp (ρ). We have another projection W 0 (ξ) → {J ⊂ S 0 |π : J → S}. Both of these projections are usually bijections, so that |Wp (ρ)| ∼ 2f . (Note that, in the above, if we replace the character ξ by its conjugate under D/D0 , this simply replaces J by its complement in S 0 .) Choose some τ00 ∈ S 0 and set τi0 = τ00 ◦ Frobi` and τi = π(τi0 ). Then we have S = {τi | i ∈ Z/f Z} and S 0 = {τi0 | i ∈ Z/2f Z}. Denote the fundamental characters for our fixed τ0 and τ00 by ω = ωτ0 and ω 0 = ωτ00 . Then we can write the other i
i
fundamental characters in terms of these two, namely, ωτi = ω ` and ωτi0 = (ω 0 )` . We also have ω = (ω 0 )`
f +1
. Note that ξ|I = (ω 0 )n for some n mod `2f − 1 and ρ|D
irreducible implies that (`f + 1) does not divide n. The following proposition, which is a combination of Propositions 3.1 and 3.2 in [BDJ], analyzes the combinatorics of the situation to make more precise the notion that |Wp (ρ)| is close to 2f . Proposition 3.3.1. Suppose ` is an odd prime and ξ, W 0 (ξ) and n are defined as above. Define A to be the set of congruence classes
mod `f + 1 of the form
−1 + (` + 1) P
if f is even
(` + 1) P
if f is odd,
i i i∈B ∗ (−1) `
i i i∈B ∗ (−1) `
27
where, in both cases, B ∗ runs over all non-empty proper subsets of {0, 1, . . . , f − 1}. The conjugacy classes of A are distinct and non-zero. Furthermore, we have
|W 0 (ξ)| =
2f
n 6∈ A
2f − 1 n ∈ A.
On the other hand, suppose ` = 2. Then 2f − 1 |W 0 (ξ)| = 2f 2f − 3
if f is even if f is odd and 3 - n if f is odd and 3 | n.
We review part of the proof as it will be useful in computing the predicted weights. Define n0a,~b,B = a(`f + 1) +
X
bi `i +
i∈B
X
bi `f +i
mod `2f − 1,
i6∈B
where a ∈ Z/(`f − 1)Z, ~b = (b0 , . . . , bf −1 ) with 1 ≤ bi ≤ ` and B ⊂ {0, . . . , f − 1}. We then have a bijection of sets W 0 (ξ) ↔ {(a, ~b, B) | n ≡ n0a,~b,B
mod `2f − 1}.
For each subset B of {0, . . . , f − 1}, there is a bijection between such triples (a, ~b, B) and solutions of the congruences n≡
X i∈B
bi `i −
X
bi `i
mod `f + 1,
(3.1)
i6∈B
with all the bi in {1, . . . , `}. But the values of the expression on the right hand side, P P i i f 0 f 0 i∈B bi ` − i6∈B bi ` , consist of the ` consecutive integers from nB − ` to nB − 1,
28
where n0B =
X
`i+1 −
i∈B
X
`i + 1.
i6∈B
So long as n 6≡ n0B mod `f + 1, we get a unique solution to the congruence (3.1) and so we have a bijection of sets mod `f + 1}.
W 0 (ξ) ↔ {B | n 6≡ n0B ∼
Also, the projection W 0 (ξ) → {J ⊂ S 0 | π : J → S} is injective. Buzzard, Diamond and Jarvis also give the following proposition, which gives criteria for determining when multiple B occur with the same (a, ~b). Proposition 3.3.2. (Proposition 3.3 in [BDJ]) The projection map from W 0 (ξ) to Wp (ρ) fails to be injective if and only if `r n ≡ m mod `f + 1 for some integers r, m with |m| ≤ `(`f −2 − 1)/(` − 1). Now consider the case where ρ|GKp is reducible. We write à ρ|GKp ∼
χ1
∗
0
χ2
! .
We define a set W 0 (χ1 , χ2 ), depending on these two characters, as follows: ( W 0 (χ1 , χ2 ) =
(V~a,~b , J) : J ⊂ S, χ1 |IKp =
Y τ ∈S
ωτaτ
Y
ωτbτ , χ2 |IKp =
τ ∈J
Y
ωτaτ
τ ∈S
Y
) ωτbτ
.
τ 6∈J
The weight set Wp (ρ) will be defined as a subset of the projection π1 (W 0 (χ1 , χ2 )) onto ! Ã χ1 ∗ , let cρ be the corresponding class the first component. Writing ρ|D ∼ 0 χ2 0 in H 1 (Kp , χ1 χ−1 2 ). To a given pair α = (V~a,~b , J) ∈ W (χ1 , χ2 ), Buzzard, Diamond ¯ ` (χ1 χ−1 )) and then define the and Jarvis associate a certain subspace Lα ⊂ H 1 (Kp , F 2
weight set by: Wp (ρ) = {V~a,~b : cρ ∈ Lα for some α = (V~a,~b , J) ∈ W 0 (χ1 , χ2 )}. 29
See [BDJ] for the definition of the subspace Lα . We analyze the set W 0 (χ1 , χ2 ) combinatorally, similar to the analysis in the irreducible case above. We will see that the projection π2 : W 0 (χ1 , χ2 ) → {J ⊂ S} is usually a bijection (and otherwise not far off) so that |W 0 (χ1 , χ2 )| ∼ 2f . Note also that interchanging χ1 and χ2 in W 0 (χ1 , χ2 ) replaces J by its complement. Find n1 and n2 in Z/(`f − 1)Z such that χ1 = ω n1 , and χ2 = ω n2 , and set n = n1 − n2 . The following three propositions are Propositions 3.4, 3.5 and 3.6 in [BDJ]. Proposition 3.3.3. Suppose that ` > 3. Define A to be the set of congruence classes mod `f − 1 of the form −1 + (` + 1) P ∗ (−1)i `i i∈B P (` + 1) (−1)i `i
if f is odd if f is even,
i∈B ∗
where, in the first case B ∗ runs over all subsets of {0, 1, . . . , f − 1}, and in the second case B ∗ runs over all non-empty proper subsets of {0, 1, . . . , f − 1}. The elements of A are distinct and non-zero. Furthermore, we have 2f + 2 |W 0 (χ1 , χ2 )| = 2f + 1 f 2
if n = 0 and f is even, if n ∈ A, otherwise.
Proposition 3.3.4. Suppose that ` = 3. Define A to be the set of congruence classes mod (3f − 1)/2 of the form −1 + 4 P ∗ (−1)i 3i i∈B 4 P (−1)i 3i i∈B ∗
30
if f is odd if f is even,
where, in the first case B ∗ runs over all subsets of {0, 1, . . . , f − 1} other than {0, 2, . . . , f − 1} and {1, 3, . . . , f − 2}, and in the second case B ∗ runs over all nonempty proper subsets of {0, 1, . . . , f −1} other than {0, 2, . . . , f −2} and {1, 3, . . . , f − 1}. The elements of A are distinct and non-zero. Furthermore, we have 2f + 2 |W 0 (χ1 , χ2 )| = 2f + 1 2f
if n = 0 and f is even, or n = (3f − 1)/2 if n ∈ A, otherwise.
Proposition 3.3.5. Suppose ` = 2. Then 2f 2f |W 0 (χ1 , χ2 )| = 2f 2f 2f
+4
if n = 0 and f is even,
+3
if n 6= 0, 3 | n and f is even,
+2
if n = 0 and f is odd,
+1
if n 6= 0 and f is odd, if 3 - n and f is even.
We recall part of the proof of the above propositions, as we will use the techniques to compute the weights in the reducible case. Define n0a,~b,B = a +
X
bi `i
mod `f − 1
i∈B
for a ∈ Z/(`f − 1)Z, ~b = (b0 , . . . , bf −1 ), with 1 ≤ bi ≤ `, and B ⊂ {0, . . . , f − 1}. ¯ denote the complement of B in {0, . . . , f − 1}. We look for triples (a, ~b, B) as Let B above such that n1 ≡ n0a,~b,B
mod (`f − 1) and
n2 ≡ n0a,~b,B¯
mod (`f − 1). 31
Such triples are in bijection with W 0 (χ1 , χ2 ). We can instead look for solutions of n≡
X
bi `i −
i∈B
X
bi `i
mod `f − 1.
i6∈B
For a particular subset B ⊂ {0, . . . f − 1}, there is a unique triple (a, ~b, B) for each solution of the above. We can use nB =
X
`i+1 −
i∈B
X
`i
i6∈B
to determine the number of solutions. In particular the values of
P
i i∈B bi ` −
P
i i6∈B bi `
are precisely the `f consecutive integers from nB + 1 − `f to nB . Thus if n 6≡ nB mod `f − 1, then we have a unique solution but if n ≡ nB mod `f − 1, then we have two solutions. Buzzard, Diamond and Jarvis also give the following proposition, which gives a criterion for determining when multiple B occur with the same (a, ~b). Proposition 3.3.6. (Proposition 3.7 in [BDJ]) The projection map from W 0 (χ1 , χ2 ) onto its first component fails to be injective if and only if `r n ≡ m mod `f − 1 for some integers r, m with |m| ≤ max{0, `(`f −2 − 1)/(` − 1)}. Note in particular that if n = 0 (e.g., in the case where ρ restricted to the decomposition group is trivial), then the projection map in the above proposition is not injective.
3.4
Cohomological
mod ` Forms Over K
For the purposes of this thesis, we will define modular forms over an imaginary quadratic field K cohomologically. Let OK denote the ring of integers of K. Analogous to the classical case, we define
32
the group Γ(n) for an ideal n ⊂ OK by (Ã Γ(n) =
a b
!
à ∈ GL2 (OK ) |
c d
a b
!
à ≡
c d
1 0
!
) mod n .
0 1
A congruence subgroup Γ is a subgroup of GL2 (OK ) which contains Γ(n) for some ideal n. The following two congruence subgroups are of particular importance: (Ã Γ0 (n) =
a b
!
à ∈ GL2 (OK ) |
c d
a b
!
à ≡
c d
∗ ∗
!
) mod n
0 ∗
and (Ã Γ1 (n) =
a b c d
!
à ∈ GL2 (OK ) |
a b c d
!
à ≡
∗ ∗ 0 1
!
) mod n .
Our definition of modular forms involves certain operators Tp called Hecke operators, which we will define in Section 5.5. Definition 3.4.1. We define a cohomological mod ` form of level n and Serre weight V to be a non-trivial cohomology class v ∈ H 2 (Γ, V ) which is a simultaneous eigenvector for the Hecke operators Tp for all primes p such that p - `n. Here, Γ is a congruence subgroup of level n.
3.5
Serre’s Conjecture for Imaginary Quadratic Fields
We need to define what it means for a representation ρ to be modular of weight ¯ ` -representation of G = V . Here, V is a Serre weight, i.e., V is an irreducible F GL2 (O/`O). We have seen that such representations are of the form V =
O p|`
33
Vp
where Vp =
O¡
¢ ¯ `. detaτ ⊗kp Symbτ −1 kp2 ⊗τ F
τ ∈Sp
To a modular form f we associate a homomorphism, which we will again denote ¯ ` [Tr : r - `n], where Tr by f , as follows. We first define the Hecke algebra, T = F is the Hecke operator for r prime. Then we can consider a modular form f to be a homomorphism ¯ ` [ar : r - `n] f :T→F Tr 7→ ar , where ar is the eigenvalue of the form f for the Hecke operator Tr . ¯ ` ), we can associate a maximal ideal To a Galois representation ρ : GK → GL2 (F in the Hecke algebra T as follows. From ρ, we get a map ¯` T→F Tr 7→ tr(ρ(Frobr )). We then get an exact sequence ¯ `, 0 → mρ → T → F with mρ = hTr − tr(ρ(Frobr ))i ⊂ T the maximal ideal we associate to ρ. We also define MΓ,V = H 2 (Γ, V ). ¯ ` ) be a continuous, irreducible representation Definition 3.5.1. Let ρ : GK → GL2 (F and V a Serre weight. We say that ρ is modular of weight V if MΓ,V [mρ ] 6= 0. Note that this is equivalent to requiring that there be a non-zero modular form ¯ ` for all f ∈ MΓ,V such that the eigenvalue ar of Tr (f ) is equal to tr(ρ(Frobr )) in F primes r - `n. The main question concerning this thesis is the following. Question 3.5.2. Let K be an imaginary quadratic field and suppose ¯ `) ρ : GK → GL2 (F 34
is a continuous, irreducible representation. Is it true that ρ is modular of weight V for every Serre weight V in the BDJ conjectural weight set W (ρ)? Remark 3.5.3. In Serre’s original conjecture, he requires that the representation ρ be odd. For a representation of GK where K is an imaginary quadratic field, there is no odd/even distinction.
35
Examples of Galois Representations In this chapter, we compute examples of Galois representations. The examples we compute come from three sources: elliptic curves, class field theory and representations arising from polynomials. Much of the data in this chapter was computed with a variety of mathematical software systems, including KANT/KASH [DF+ 97], Magma [BCP97], PARI/GP [The05] and Sage [Ste].
4.1
Examples from Elliptic Curves
Fix a prime ` and let E be an elliptic curve over K = Q(i) which is supersingular at ` and which has good reduction at `. To such an elliptic curve E one can associate a mod ` Galois representation ρE such that 1. the level of ρE is equal to the conductor of the elliptic curve E; 2. the character of ρE is trivial; and 3. the traces ap of ρE (Frobp ) are given by the sequence {ap }E associated to the elliptic curve, reduced mod `. Furthermore, with the above assumptions, we know the local behaviour of ρE at `. In particular, we will look at the case ` = 7. Letting I7 denote the inertia group at 7, we have
à ρ E |I 7 ∼
ω
0
0 ω7
! ,
where ω denotes the level 2 fundamental character. Using the BDJ recipe, we have χ1 = ω 1 χ2 = ω 7 ,
36
so n1 = 1 and n2 = 7. The inertia degree for 7 in K = Q(i) is f = 2, so we will have a ∈ Z/(`f − 1)Z = Z/48Z ~b = (b0 , b1 ) and B ⊂ {0, 1}. We need to find triples (a, ~b, B) such that n1 ≡ na,~b,B = a +
X
7i bi
mod 48,
7i bi
mod 48.
and
i∈B
n2 ≡ na,~b,B¯ = a +
X ¯ i∈B
For each of the four subsets B, we get one solution (a, ~b). We then compute ~a = (a0 , a1 ) by computing a ≡ a0 + 7a1 mod 48. We list the predicted weights in Table 4.2. Table 4.1: Weights for elliptic curves mod 7 ` char ~a = (a0 , a1 ) ~b = (b0 , b1 ) 7 7 7 7
1 1 1 1
(0, 0) (0, 0) (1, 0) (0, 1)
(1, 1) (7, 7) (5, 7) (7, 5)
The examples we compute in this section come from elliptic curves over Q(i) computed by Cremona in [Cre84]. Cremona computed modular forms (with coefficients in Q, i.e. with trivial weight) corresponding to these elliptic curves, so we already know that the representations associated to these elliptic curves are modular. Here we compute all the weights in which we expect to find modular forms giving rise to these representations. By computing the modular forms for those weights and levels, we provide computational evidence for the BDJ conjecture in this setting.
37
In his tables, Cremona lists elliptic curves by giving their ai coefficients for the Weierstrass equation E : y 2 + a1 xy + a3 y = x3 + a2 x2 + a4 x + a6 . In order to determine whether these elliptic curves are supersingular and have good reduction at `, we first complete the square to get an equation of the form E : y 2 = 4x3 + b2 x2 + 2b4 x + b6 where b2 = a21 + 4a2 b4 = 2a4 + a1 a3 b6 = a23 + 4a6 . We will also need b8 = a21 a6 + 4a2 a6 − a1 a3 a4 + a2 a23 − a24 , and ∆ = −b22 b8 − 8b34 − 27b26 + 9b2 b4 b6 . This last quantity ∆ is called the discriminant of the Weierstrass equation. Considering ∆ modulo ` tells us whether E has good reduction modulo `, i.e., whether the reduction of E modulo ` is non-singular. To test whether E is supersingular at `, we use the following theorem. (See, e.g., [Sil86, p. 140].) Theorem 4.1.1. Suppose K is a finite field of characteristic ` 6= 2. Let E be an elliptic curve over K given by Weierstrass equation E : y 2 = f (x), ¯ The curve E is superwhere f (x) ∈ K[x] has degree 3 and has distinct roots in K. singular if and only if the x`−1 term in f (x)(`−1)/2 has coefficient equal to zero.
38
Of the curves listed in Cremona’s table, we found several which are both supersingular and have good reduction at ` = 7. We list the ai coefficients for these curves in the following table, along with their conductors f and the norms of the conductors N f. We indicate those curves whose sequences {ap } suggest that the corresponding modular form is actually Galois over Q (judging by whether ap equals a¯p for split primes pOK = pp¯). We are, of course, more interested in those which are not Galois over Q. In the rightmost column, we indicate whether the corresponding modular form was found for all of the predicted weights (listed above in Table 4.1). Table 4.2: Elliptic curves over Q(i) mod 7 with several levels a1 1+i i 1 i
a2 0 1−i 1 −1
a3 1+i i 1 0
a4 a6 1−i 0 −i 0 0 0 2−i i
f Nf (6 + 6i) 72 (9 + 7i) 130 (15) 225 (19 + 9i) 442
Gal/Q X X
f X X X X
In Section 6.3, we give the {ap }E sequence for some small primes p and the {ap } coefficients computed for the corresponding modular forms.
4.2
Examples from Polynomials
In this section we will examine examples of Galois representations arising from polynomials. We will determine the level, character and predicted weights for each representation.
4.2.1 Techniques & Background Here we will review some background, stating basic theorems and describing general techniques, which will be used in the subsequent sections to analyze the various examples of Galois representations. Let p(x) be an irreducible polynomial in Q[x], and denote by L the Galois closure of the field generated by p(x) over Q. We will look at 2-dimensional 39
mod `
¯ representations of the Galois group Gal(L/Q). Taking the image of Gal(Q/L) to be trivial, we get a representation ¯ ` ), ρQ : GQ → GL2 (F which factors through Gal(L/Q). We are interested in representations of GK for imaginary quadratic fields K, so we take the restriction to K of the above representation, giving us ¯ ` ), ρK : GK → GL2 (F which factors through Gal(M/K), where M is the composite field LK. If we start with a representation ρQ over Q which is odd, i.e. det ρ(σ∞ ) = −1, then we already know ρQ is modular by Serre’s conjecture. Instead, we will start with even representations ρQ , and look at the base change to K. Once we have a representation ¯ ` ), ρK : GK → GL2 (F we need to compute the level, weight and character of ρK in order to figure out where to look for corresponding modular forms. We will of course also have to compute the coefficients {ap } for primes p. To compute the level of ρK , we will look at all primes dividing the discriminant of the polynomial, except `. For each such prime p, we compute n(p, ρ|Q ). In the examples we compute we will have p unramified in K for all primes p dividing the level N of the representation ρ|Q . When this is the case, we have n(p, ρ|K ) = n(p, ρ|Q ) for primes p lying above p. The corresponding level to check will be n=
Y
pn(p,ρ) .
As discussed earlier, we know that for any prime p which is tamely ramified in L, we have n(p, ρ) = dim(V /V0 ), where V0 is the subspace of V fixed by the inertia group IP for some prime P of L lying above p. We often make use of the basic fact that |IP |
40
is equal to the ramification index eP/p . If p is wildly ramified in L, the determination of n(p, ρ) requires further analysis. To determine the possible Serre weights V~a,~b , we consider the primes p lying above `. For each such prime p, we need to consider the local behaviour of ρK at p. We then apply the weight recipe in [BDJ] to compute the possible weights Vp . There are two cases: when the restriction of ρK to the decomposition group Dp is irreducible and when it is reducible. In either case, we will examine the restriction of ρK to the inertia group Ip . We make use of the combinatorial analysis in [BDJ], which provides formulas for the number of different weights Vp for each prime p above `, to check that we have found all possible weights. In those cases where we compute the base change from a representation over Q, we make use of some basic facts regarding base change to compute the coefficients {ap }. Recall that we want to compute the restriction of the representation ρ to the imaginary quadratic field K = Q(i), ¯ ` ). ρK = ρ|K : GK → GL2 (F Let M = KL be the composite of K and L. Since both K and L are Galois over Q, we know that M is Galois over Q and M is Galois over K. In our examples, we will consider totally real extensions L, so we have K ∩ L = Q, and thus G = Gal(M/K) ∼ = Gal(L/Q) and
Gal(M/Q) ∼ = Gal(K/Q) × Gal(L/Q) σ 7→ (σ|K , σ|L ).
Suppose p is a rational prime which is unramified in M , and let P be a prime of M lying above p. Let pK = P ∩ K and, likewise, pL = P ∩ L. We need to compute apK for M over K from ap for L over Q. We examine the two cases: p splits in K and p is inert in K. Case 1 (p splits in K): We have p = pK p¯K . Suppose for some prime P of M
41
lying over p, we have pK = P ∩ K. The inertia degree f (pK /p) is equal to 1. Thus we have
·
¸ µ· ¸ · ¸¶ M/K K/Q L/Q = × . P pK pL
The decomposition group DpK is trivial. Then, since FrobpK ∈ DpK , we must have ·
We now have
¸ K/Q = FrobpK = 1. pK
µ· ρK
M/K P
µ·
¸¶ =ρ
L/Q pL
¸¶
and so, of course, the traces will be the same as well, i.e., apK = a¯pK = ap . Case 2 (p is inert in K): We have p = p2K . The inertia degree f (pK /p) is equal to 2. Thus we have
and so
·
µ· ρK
¸ µ· ¸ · ¸¶2 M/K K/Q L/Q = × P pK pL ÷ ¸2 · ¸2 ! L/Q K/Q × = pK pL à · ¸2 ! L/Q = id × , pL
M/K P
÷
¸¶ =ρ
L/Q pL
¸2 !
µ· 2
=ρ
L/Q pL
¸¶ .
We can use this relationship to compute ap from ap . Alternatively, we could use the following method. Denote by ap the trace of ρQ (Frobp ) and denote by bp the trace of ρK (FrobpK ). ¯ ` with αβ = det(ρQ (Frobp )). Then we have Suppose ap = α + β, for some α, β ∈ F bp = α 2 + β 2 = (α + β)2 − 2 det(ρQ (Frobp )) = a2p − 2 det(ρQ (Frobp )). We want to compute the sequence {ap } associated to the representation ρ, i.e. 42
we want to compute tr(ρ(Frobp )) for each prime p of OK where p is unramified in ¯K the residue field of M = LK. Let P be a prime of M lying over p. Denote by O OK and likewise for M . Since p is unramified in M , the Frobenius automorphism Frobp is uniquely defined ¯M /O ¯K ) of the residue ([Jan96, p 125]). Also, Frobp generates the Galois group Gal(O field extension corresponding to the field extension M over K. We know that ¯M /O ¯K ) ∼ Gal(O = Gal(Fqf /Fq ), ¯K | and f = f (P/p) is the inertia degree of P over p. Thus the order of where q = |O the element Frobp is equal to the inertia degree f . When the image ρ(Gal(M/K)) is isomorphic to Gal(M/K), we know that the order of ρ(Frobp ) is equal to the order of the Frobp . Often, the order of ρ(Frobp ) is sufficient to determine the trace of ρ(Frobp ).
4.2.2 Dihedral Group D4 In this section, we compute an example with dihedral group D4 , which appears in Ash, Doud and Pollack [ADP02]. Example 4.2.1. Representation
mod 5 with level n = 29
The totally real polynomial x4 − x3 − 3x2 + x + 1,
disc = 52 × 29,
has Galois closure L defined by the polynomial x8 − 6x7 − 13x6 + 102x5 + 27x4 − 438x3 + 25x2 + 252x + 49. The Galois group G = Gal(L/Q) is isomorphic to the dihedral group D4 . We will take ` = 5. There is one irreducible 2-dimensional mod 5 representation of D4 . Since the extension L over Q is totally real, this representation is necessarily even and hence not covered by Serre’s conjecture. (However, since it has solvable image, it is known
43
to be modular by proven cases of Artin’s conjecture.) We consider the base change ρK from Q to K = Q(i) of this representation. In the following table we give, for each conjugacy class in the image of ρ, a representative element along with its order, length (of conjugacy class), determinant, ap = trace, and bp = tr2 − 2 det. For a split prime pOK = pp¯ we have tr(ρ(Frobp )) = ap = a¯p and for an inert prime pOK = p we have tr(ρ(Frobp )) = bp . Ã Ã Ã Ã Ã
rep 1 0 0 1 4 0 0 4 0 1 1 0 4 0 0 1 0 4 1 0
!
order
length
det ap
bp
1
1
1
2
2
2
1
1
3
2
2
2
4
0
2
2
2
4
0
2
4
2
1
0
3
! ! ! !
From the above table, we see that det(ρ) is a quadratic character, and so we get a nontrivial character for ρ, which is a quadratic character on (OK /29OK )∗ . We will denote this character by ε29 . To compute the level, we find the ramification index at p = 29 is e29 = 2. Since p = 29 is tamely ramified, we have n(29, ρ) = dim(V /V0 ). The inertia group I29 has order 2 and is not central. The representation ρ restricted to such a subgroup of D4 fixes a space of dimension 1, so n(29, ρ) = 1 and the level of this representation is n = 29. We use the BDJ recipe to compute the weights for ` = 5. We compute the ramification index e5 = 2 and the inertia degree f5 = 2. Furthermore, we compute that ρ|I5 fixes a one dimensional subspace and so is of the form à ρ|I5 ∼
ω2 0 0
44
1
! ,
where ω is the mod 5 cyclotomic character. We use the reducible case of BDJ, and we have χ1 = ω 2 χ2 = ω 0 , so n1 = 2 and n2 = 0. The inertia degree for 5 in K = Q(i) is f = 1, so we will have a ∈ Z/(`f − 1)Z = Z/4Z ~b = (b0 ) and B ⊂ {0}. We need to find triples (a, b0 , B) such that n1 ≡ na,b0 ,B = a +
X
5i bi
mod 4,
5i bi
mod 4.
and
i∈B
n2 ≡ na,b0 ,B¯ = a +
X ¯ i∈B
For each of the two subsets B, we get one solution (a, b0 ). In particular, for B = {0}, we get a = 0, b0 = 2 and for B = ∅, we get a = 2, b0 = 2. The analysis is the same regardless of which prime we choose above ` = 5, and so, for K = Q(i), we combine the results to get the various possible weights, which we list in Table 4.3. Table 4.3: D4 representation mod 5 with level n = 29 ` level char ~a = (a0 , a1 ) ~b = (b0 , b1 ) f 5 29 ε29 (0, 0) (2, 2) X 5 29 ε29 (0, 2) (2, 2) 5 29 ε29 (2, 0) (2, 2) 5 29 ε29 (2, 2) (2, 2) X
For two of the four weights in the above table, we did not find the form as expected. There was a form found in those weights that looks as though it might be a twist of the expected form. Further investigation is required to determine what is happening here. In Section 6.4.1 we provide a table with the orders of Frobp along with the coeffi45
cients ap of the corresponding system of eigenvalues for some small primes p for this representation.
4.2.3 Alternating Group A4 Example 4.2.2. Representation
mod 3 with level n = 61
In [Fig99, p 117], Figueiredo gives three examples of A4 representations. The first of these examples comes from the polynomial x4 − 7x2 − 3x + 1
disc = 32 × 612 .
Figueiredo considers the mod 3 representation arising from this polynomial using the isomorphism A4 ∼ = PSL2 (F3 ). He shows that there must be a lift of this representation ¯ 3 ). I will instead compute representations directly from the Aˆ4 extension, to GL2 (F where Aˆ4 is a double cover of A4 , isomorphic to SL2 (F3 ). From the database of Kl¨ uners and Malle [KM], we find the polynomial giving the Aˆ4 extension of Figueiredo’s polynomial: x8 + 3x7 − 11x6 − 9x5 + 21x4 + 9x3 − 11x2 − 3x + 1 which has Galois closure given by the following degree 24 polynomial: x24 − 9x23 − 206x22 + 1680x21 + 16053x20 − 112863x19 − 585638x18 + 3495552x17 + 9763561x16 − 57615780x15 − 68523951x14 + 519956256x13 + 95882805x12 − 2594932704x11 + 1213595253x10 + 6932798424x9 − 6682968027x8 − 8573027148x7 + 13459018536x6 + 1679228685x5 − 10467091917x4 + 4152416400x3 + 1245998376x2 − 1023909417x + 155570139 We denote this Galois closure by L.
46
We take ` = 3. There is only one irreducible 2-dimensional mod 3 representation ¯ 3) ρ : GK → GL2 (F factoring through Gal(LK/K). We get this representation by taking the base change to K = Q(i) from the representation ρQ of GQ , which we get simply by restricting GQ to Gal(L/Q) and then applying the following isomorphisms and inclusion: Gal(L/Q) ∼ = Aˆ4 ∼ = SL2 (F3 ) ,→ GL2 (F3 ). For the level, we need only compute n(61, ρ). We have the ramification index e61 = 3 in L, and so 61 is tamely ramified and n(61, ρ) = dim(V /V0 ). We check that ρ restricted to the subgroup of order 3 fixes a 1-dimensional subspace of V , so n(61, ρ) = 1 and the level of ρ is n = 61. Since the image of ρ is SL2 (F3 ), we see that det(ρ) is trivial, and so the character of ρ is trivial as well. To compute the weights, we look at the representation locally at ` = 3 and apply the BDJ recipe. We compute the ramification index e3 = 4 and the inertia degree f3 = 2, and we note that the inertia degree of 3 in K = Q(i) is f = 2. The decomposition group at 3 for L over Q has order 8, and the only order 8 subgroup is the quaternion group Q8 . If we consider the restriction of ρQ to the decomposition group, we would be in the irreducible case of BDJ. However, we want to consider the restriction of ρQ first to K = Q(i), which we denote by ρ, and then to the decomposition group at 3. Then we have ρ restricted to D3 is reducible, and we can write
à ρ|D3 ∼
ω (`
2 −1)/4
and so we have χ1 = ω (`
0 ω −(`
0 2 −1)/4
2 −1)/4
= ω2
2 −1)/4
χ2 = ω −(`
!
= ω −2 .
Thus n1 = 2 and n2 = 6. (Recall that nν ∈ Z/(`f − 1)Z = Z/8Z.) We need to find 47
triples (a, ~b, B) with a ∈ Z/8Z B ⊂ {0, . . . , f − 1} and ~b = (b0 , b1 ), with 1 ≤ bi ≤ ` = 3 such that n1 ≡ na,~b,B = a +
X
3i bi
mod 8,
3i bi
mod 8.
and
i∈B
n2 ≡ na,~b,B¯ = a +
X ¯ i∈B
For two of the four subsets B (namely, B = {0} and B = {1}) we get one solution (a, ~b), while for the other two (B = {0, 1} and B = ∅) we get two solutions (a, ~b). We then compute ~a = (a0 , a1 ) by writing a = a0 + 3a1 mod 8. In Table 4.4 we list the possible weights. Table 4.4: A4 representation mod 3 with level n = 61 ` level B ~a = (a0 , a1 ) ~b = (b0 , b1 ) f 3 61 {0, 1} (0, 2) (1, 1) X 3 61 {0, 1} (0, 2) (3, 3) X 3 61 {1} (1, 1) (2, 2) X 3 61 {0} (0, 0) (2, 2) X 3 61 ∅ (2, 0) (1, 1) X 3 61 ∅ (2, 0) (3, 3) X
We need to compute the coefficients {ap }. When p splits in K, say p · OK = pp¯, we have ap = a¯p = ap . The image of our representation is SL2 (F3 ), so we have det = 1. Therefore if p = p is inert in K, we get bp = tr(ρ(Frobp ))2 − 2 · det(ρ(Frobp )) = b2p − 2 ≡ b2p + 1
mod 3.
In the following table we give representatives for each conjugacy class in SL2 (F3 ), 48
the order of those elements, the trace (equal to ap when p splits in K) and bp when p is inert in K. Rep
à à à à à à Ã
1 0
!
0 1 −1
0
0
−1
0
1
−1 −1 0 −1 1 −1 0 −1 1
0
0 −1 1 0
1 1
−1 1
Order
tr = ap
bp
1
−1
−1
2
1
−1
3
−1
−1
3
−1
−1
4
0
1
6
1
−1
6
1
−1
! ! ! ! ! !
In Section 6.4.2 we provide a table with the orders of Frobp along with the coefficients of the corresponding systems of eigenvalues for some small primes p for this representation. Example 4.2.3. Representation
mod 3, level n = 79
Figueiredo’s second example in [Fig99] comes from the polynomial P = x4 − x3 − 24x2 + x + 11
disc = 34 × 792 .
This is a totally real A4 extension of Q. It is a subfield of an Aˆ4 extension generated by the following polynomial x8 − 17x6 + 60x4 − 29x2 + 1,
49
which has as its Galois closure the field defined by the following degree 24 polynomial x24 − 255x22 + 27807x20 − 1706560x18 + 65353749x16 − 1637285385x14 + 27343876078x12 − 303898414365x10 + 2196602852661x8 − 9790619278320x6 + 24093731990727x4 − 25093789540875x2 + 2600398130625. Let L denote the field generated by this degree 24 extension. We take ` = 3, so we will be considering mod 3 representations of Gal(L/Q) ∼ = Aˆ4 ∼ = SL2 (F3 ). There is only one such irreducible degree 2 representation. We need to compute the level of this representation. Since we are considering this representation mod 3 and the only primes ramifying in this extension are 3 and 79, we need only consider p = 79 for the level. The ramification at p = 79 is tame and so n(79, ρ) = dim(V /V0 ). We compute the ramification index e79 = 3 and denote the inertia group at 79 by I79 . We find that ρ|I79 fixes a 1-dimensional subspace of V , so n(79, ρ) = 1 and the level of this representation is n = 79. Since the image of this representation is isomorphic to SL2 (F3 ), the character will be trivial. We now use the BDJ recipe to compute the weights. For ` = 3 we find the ramification index e3 = 3 and the inertia degree f3 = 2. Globally, the decomposition group at 3 would then have order 6. However, as in the previous example, we consider the restriction of the representation to K = Q(i), and so the decomposition group D3 in this case will have order 3. We are thus in the reducible case of BDJ. We can write
à ρ|D3 ∼
1 ∗ 0 1
and so we have χ1 = ω 0 χ2 = ω 0 ,
50
! ,
with n1 = n2 = 0 (and so also n = 0). We need to find triples (a, ~b, B) with a ∈ Z/8Z, ~b = (b0 , b1 ) with 1 ≤ bi ≤ ` = 3 and B ⊂ {0, 1} such that 0 ≡ na,~b,B = a +
X
3i bi
mod 8,
3i bi
mod 8.
and
i∈B
0 ≡ na,~b,B¯ = a +
X ¯ i∈B
For two of the four subsets B (namely, B = {0, 1} and B = ∅) we get one solution ~b (but two solutions for (~a, ~b)), while for the other two (B = {0} and B = {1}) we get two solutions (a, ~b) each. However there is some symmetry here: we get the same solutions for B = {0} as for B = {1}, and we get the same solutions for B = ∅ as for B = {0, 1}. This symmetry is expected by Proposition 3.3.6, since n = 0. We compute ~a = (a0 , a1 ) by writing a = a0 + 3a1 mod 8. In Table 4.5 we list the possible weights. Table 4.5: A4 representation mod 3 with level n = 79 ` level char B ~a = (a0 , a1 ) ~b = (b0 , b1 ) f 3 3 3
79 79 79
1 1 1
{0, 1}, ∅ {1}, {0} {1}, {0}
(0, 0) (1, 2) (2, 1)
(2, 2) (1, 3) (3, 1)
X
The weights in Table 4.5 are the weights in W 0 (χ1 , χ2 ) in the BDJ notation for the reducible case. Recall that, for these weights to be in the predicted weight set Wp (ρ), they must satisfy the additional condition that the corresponding cohomology ¯ ` (χ1 χ−1 )). In this case, one class cρ be contained in a certain subspace Lα ⊂ H 1 (Kp , F 2
can show that this condition is not met for the latter weights (those with ~b = (1, 3) and ~b = (3, 1)). Thus we only expect this form to show up in the former weight (with ~b = (2, 2)) and, as indicated in the final column, that is indeed what we found. In Section 6.4.2 we provide a table with the orders of Frobp along with the coef51
ficients of the corresponding systems of eigenvalues for some small primes p for this representation.
4.3
Examples from Class Field Theory
4.3.1 Dihedral Group D3 For this section, we assume ` 6= 3. Write the elements of D3 as rotations rk and ¯ ∗ , we get one irreducible reflections srk for 0 ≤ k ≤ 2. If w is a third root of unity in F `
2-dimensional representation ρ of D3 . We can write the action of ρ on the elements of D3 as follows (see, e.g., [Ser77, p.36]) Ã ρ(rk ) =
wk 0
!
0 w
à ρ(srk ) =
,
−k
0
w−k
wk
0
! .
In the following table, we give the trace, determinant and order of the images of ρ on the elements of D3 . r0
r1
r2
sr0
sr1
sr2
trace
2
−1 −1
0
0
0
det
1
1
1
−1
−1
−1
order
1
3
3
2
2
2
To compute the coefficients bp in the base change to K = Q(i) when p is inert, we use the formula bp = tr(ρQ (Frobp ))2 − 2 · det(ρQ (Frobp )), where p is the rational prime below p. We get the following values for bp when p is inert in Q(i).
bp
r0
r1
2
−1 −1
Example 4.3.1. Representations
r2
sr0
sr1
sr2
2
2
2
mod 5 and
13 + 28i and n = 8 + 35i 52
mod 7 with levels n = 8 + 17i, n =
The following examples are not arising as base change of even representations over Q, but are representations directly over K = Q(i). We get these examples by considering quadratic extensions of Q(i) which are ramified only at a single prime p, split over Q, and which have class group isomorphic to the cyclic group of order 3. I would like to thank Gabor Wiese for showing me this source of examples. At the time, we had been considering these examples
mod 2, but as 2 is ramified in
Q(i), it is not covered by the BDJ recipe. Schein’s extension of the BDJ recipe to the ramified case ([Sch]) does not cover ` = 2. In the following we will consider these representations mod 5 and mod 7. This allows us to see the different behaviour in the weights modulo an inert prime as compared to a split prime. The representation ρ will factor through an extension L of K = Q(i) where G = Gal(L/K) is isomorphic to D3 . In all cases the representation will have a ¯ ∗. quadratic character ε : (OK /p)∗ → F `
For the level, we need only consider this single ramified prime p, which will have ramification index ep = 2 in the extension L over K. The representation ρ restricted to the order 2 subgroup of D3 fixes a one-dimensional subspace, so that the level in each case will be precisely equal to this prime, i.e., n = p. In all three examples, the primes above 5 and 7 in Q(i) will be unramified in the dihedral extension L over K. Thus, in all cases, the representation ρ restricted to the inertia group I at ` will be trivial and so we have χ1 = ω 0 χ2 = ω 0 , and n = n1 = n2 = 0. The difference in the weight computations arises from the prime ` being split or inert. Note that for both cases n = 0 implies, according to Proposition 3.3.6, that we will have multiple subsets B with the same (a, ~b).
53
For ` = 5 (exemplifying the split case), we need to find triples (a, b0 , B) with a ∈ Z/4Z, 1 ≤ b0 ≤ 5 and B ⊂ {0} such that
X
0 ≡ na,b0 ,B = a +
5i bi
mod 4,
5i bi
mod 4.
and
i∈B
X
0 ≡ na,b0 ,B¯ = a +
¯ i∈B
If we take either B = {0} or B = ∅, we have the congruences 0 ≡ a + b0 0≡a
mod 4,
and
mod 4.
Thus a is 0 and b0 = 4. For ` = 7 (exemplifying the inert case), we look for triples (a, ~b, B) with a ∈ Z/48Z, ~b = (b0 , b1 ) with 1 ≤ bi ≤ 7 and B ⊂ {0, 1} such that 0 ≡ na,b0 ,B = a +
X
7i bi
mod 48,
7i bi
mod 48.
and
i∈B
0 ≡ na,b0 ,B¯ = a +
X ¯ i∈B
We again have some symmetry: the cases B = {0, 1} and B = ∅ will yield the same system of congruences and the cases B = {1} and B = {0} will yield the same system of congruences. For the former, we have 0 ≡ a + b0 + 7b1
mod 48,
0 ≡ a mod 48, 54
and
giving ~b = (6, 6) and a = 0 so ~a = (0, 0). In the latter case, we have 0 ≡ a + b0 0 ≡ a + 7b1
mod 48,
and
mod 48,
giving ~a = (5, 6) with ~b = (1, 7) and ~a = (6, 5) with ~b = (7, 1). The primes for which we found such dihedral extensions are n = (8 + 17i) (lying over p = 353), n = (13 + 28i) (lying over p = 953) and n = (8 + 35i) (lying over p = 1289). In Table 4.6, we list all the combinations of weights, level, character and prime ` for the above examples, and indicate in the final column (under f ) whether a corresponding modular form was found. Table 4.6: D3 representations mod 5 and mod 7 with several levels ` 5 7 7 7 5 7 7 7 5 7 7 7
level 8 + 17i 8 + 17i 8 + 17i 8 + 17i 13 + 28i 13 + 28i 13 + 28i 13 + 28i 8 + 35i 8 + 35i 8 + 35i 8 + 35i
char ~a = (a0 , a1 ) ~b = (b0 , b1 ) ε8+17i (0, 0) (4, 4) ε8+17i (0, 0) (6, 6) ε8+17i (5, 6) (1, 7) ε8+17i (6, 5) (7, 1) ε13+28i (0, 0) (4, 4) ε13+28i (0, 0) (6, 6) ε13+28i (5, 6) (1, 7) ε13+28i (6, 5) (7, 1) ε8+35i (0, 0) (4, 4) ε8+35i (0, 0) (6, 6) ε8+35i (5, 6) (1, 7) ε8+35i (6, 5) (7, 1)
f X X X X X X X X X X X X
In Section 6.5.1 we provide tables with the orders of Frobp along with the coefficients ap of the corresponding system of eigenvalues for some small primes p for each of the three levels n above. 55
4.3.2 Dihedral Group D5 Example 4.3.2. Representation
mod 11 with level n = 19 + 20i
In this section we look at another example which is a representation directly over K = Q(i), not a base change from a representation over Q. This one was constructed in the same way as the D3 examples of this sort in Section 4.3.1. For the prime 19 + 20i (lying over p = 761), we get a quadratic extension of Q(i) which is ramified only at 19 + 20i and which has class group isomorphic to the cyclic group of order 5. Thus we get an extension L of K = Q(i) such that the Galois group G = Gal(L/K) is isomorphic to D5 . We will consider this representation modulo ` = 11, for which prime we have an irreducible 2-dimensional representation of D5 . The image of det(ρ) is ±1, so the ¯ ∗. character of ρ will be a quadratic character ε19+20i : (OK /(19 + 20i)OK )∗ → F `
In the following table we give, for each conjugacy class in the image of ρ, a representative element along with its order, length (of conjugacy class), determinant and trace. Ã Ã Ã Ã
rep 1 0 0 1 0 1 1 0 4 0 0 3 5 0 0 9
!
order
length
det
trace
1
1
1
2
2
5
10
0
5
2
1
7
5
2
1
3
! ! !
The ramification index of p = 19 + 20i in L over K is ep = 2. The fixed space of ρ restricted to the order 2 subgroup of D5 has dimension 1 so the level of ρ is n = 19 + 20i. To compute the weights, we apply the BDJ recipe in the reducible case. The prime above 11 is unramified in the extension L over K, so the restriction of ρ to the
56
inertia group I for ` = 11 is trivial. Thus we have χ1 = ω 0 χ2 = ω 0 , so n1 = n2 = 0. Note that n = 0 implies, according to Proposition 3.3.6, that we will have multiple subsets B with the same (a, ~b). We look for triples (a, ~b, B) with a ∈ Z/120Z, ~b = (b0 , b1 ) with 1 ≤ bi ≤ 11 and B ⊂ {0, 1} such that 0 ≡ na,b0 ,B = a +
X
11i bi
mod 120,
11i bi
mod 120.
and
i∈B
0 ≡ na,b0 ,B¯ = a +
X ¯ i∈B
For the subsets B = {0, 1} and B = ∅ we get the same system of congruences, namely: 0 ≡ a + b0 + 11b1
mod 120,
and
0 ≡ a mod 120, giving ~b = (10, 10) and a = 0 so ~a = (0, 0). For the subsets B = {1} and B = {0} we get the same system of congruences, namely: 0 ≡ a + b0 0 ≡ a + 11b1
mod 120,
and
mod 120,
giving ~a = (9, 10) with ~b = (1, 11) and ~a = (10, 9) with ~b = (11, 1). In Table 4.7, we list the possible weights with the level and character for this example and indicate in the final column whether a corresponding form was found. The case for which it has not yet been found has not been checked due to size limitations in the computations.
57
Table 4.7: D5 representation mod 11 with level n = 19 + 20i ` level char ~a = (a0 , a1 ) ~b = (b0 , b1 ) f 11 11 11
19 + 20i ε19+20i 19 + 20i ε19+20i 19 + 20i ε19+20i
(0, 0) (9, 10) (10, 9)
(10, 10) (1, 11) (11, 1)
X X
In Section 6.5.2 we provide a table with the orders of Frobp along with the coefficients of the corresponding systems of eigenvalues for some small primes p for this representation.
58
Computing Modular Forms over Q(i) 5.1
Borel-Serre Duality
Recall that the space we want to compute is H 2 (Γ, V ) for some congruence subgroup Γ and some Serre weight V . Instead of computing this cohomology group directly, we use Borel-Serre duality to compute a homology group with coefficients in the Steinberg module. The homology group is computationally friendly because we can express the Steinberg module (and a free resolution of it) in terms of modular symbols. Let K be a number field with ring of integers O and let Γ be a subgroup of finite index in GL2 (O). We will assume that K has class number 1 so that O is a PID. Denote by ν the virtual cohomological dimension of Γ. Applying the formula in Ash [Ash94, p.330], we see that ν = 2r1 + 3r2 − 1, where r1 is the number of real and 2r2 the number of complex embeddings of K. Let R be a commutative ring with identity such that the torsion in Γ is invertible in R. We define the Steinberg module to be the R[Γ]-module H ν (Γ, R[Γ]) and denote it by St. Suppose M is an R[Γ]-module. Borel-Serre duality [BS73, p.482-483] gives an isomorphism ∼
H α (Γ, M ) −→ Hν−α (Γ, St ⊗ M ) for any integer α. For imaginary quadratic fields K, we have ν = 2. In our case, we want to compute H 2 (Γ, V ) for a Serre weight V , so via Borel-Serre duality, we will instead be computing H0 (Γ, St⊗V ). We will need to know how to compute the Steinberg module. Ash gives a description, which we recall here, of the Steinberg module in terms of “universal minimal modular symbols”. In [Ash94], Ash defines this space for arbitrary dimension n. Here, we restrict to the case n = 2. The space of universal minimal modular symbols is canonically isomorphic to the
59
Steinberg module as we have defined it above; from now on we will take the modular symbols description as our definition. Consider the set of formal R-linear sums of symbols [v] = [v1 , v2 ] where the vi are unimodular columns in O2 , i.e., vi = [a b]T with gcd(a, b) = 1. Mod out by the R-module generated by the following elements: 1. [v2 , v1 ] + [v1 , v2 ]; 2. [v] = [v1 , v2 ] whenever det(v) = 0; and 3. [v1 , v3 ] − [v1 , v2 ] − [v2 , v3 ], where the vi again run over all unimodular columns in O2 . This quotient module is now our definition of the Steinberg module St. We denote the image of [v] in St by [v]∗ . In order to compute the homology of Γ with coefficients in St ⊗ M , we will first give a free resolution of St · · · → S2 → S1 → S0 → St and then tensor with the R[Γ]-module M · · · → S2 ⊗ M → S1 ⊗ M → S0 ⊗ M → St ⊗ M. We will then take Γ coinvariants: · · · → (S2 ⊗ M )Γ → (S1 ⊗ M )Γ → (S0 ⊗ M )Γ → (St ⊗ M )Γ and, finally, take the homology of this chain complex. The first thing we need is an R[Γ]-free resolution of St. We now describe the resolution given by Ash in [Ash94], which is based on the resolution in Lee and Szczarba [LS76]. The R[Γ]-modules Sk of this resolution are described in a manner similar to that of the Steinberg module. Consider the set of formal R-linear sums of symbols [v] = [v1 , v2 , · · · , v2+k ], where each vi is a unimodular column in O2 . Mod out by the R[Γ]-module generated by the following elements: 60
1. [vσ(1) , . . . , vσ(2+k) ] − sgn(σ)[v1 , . . . , v2+k ]; and 2. [v] if v1 , . . . , v2+k are all contained in a hyperplane in K 2 . Here, the vi again run over all unimodular columns in O2 and σ runs over all permutations of {1, . . . , 2 + k}. We take this quotient module to be Sk in our free resolution of St: · · · → S2 → S1 → S0 → St and denote the image of [v] in Sk by [v] again. We define the boundary operator Sk → Sk−1 to be ∂[v] =
X
(−1)i [v1 , . . . , vˆi , . . . , v2+k ]
and the map S0 → St is simply [v] 7→ [v]∗ . Ash [Ash94, p.332] notes that the proof that the Sk chain complex is an R[Γ]-free resolution of St is similar to the proof given in [LS76]. We are interested in computing H 2 (Γ, M ), so (recalling that, for us, ν = 2) we apply the Borel-Serre isomorphism ∼
H 2 (Γ, M ) −→ H0 (Γ, St ⊗ M ) to see that we will instead be computing H0 (Γ, St ⊗ M ). Thus we will only be concerned with the following part of the chain complex: S1 −→ S0 −→ St. We tensor with M and then take coinvariants by Γ to get ∂
(S1 ⊗ M )Γ −→ (S0 ⊗ M )Γ −→ (St ⊗ M )Γ . We will compute the homology of degree 0 of this, i.e., H0 (Γ, St ⊗ M ) = ker ε/im∂, 61
where ∂
ε
(S1 ⊗ M )Γ −→ (S0 ⊗ M )Γ −→ 0. So we will compute (S0 ⊗ M )Γ /∂((S1 ⊗ M )Γ ), which amounts to computing (S0 ⊗M )Γ modulo the relations [v1 , v3 ]−[v1 , v2 ]−[v2 , v3 ], where the vi run over all unimodular columns in O2 . Note that, in our case, for both St and S0 , we are looking at a space of ordered pairs of unimodular columns in O2 , which is the same as paths between cusps (K ∪ {∞}) in the modular symbols methods of Cremona et al.
5.2
An Algebraic Proposition
The following proposition is analogous to Proposition 4.3 in the doctoral thesis of Martin [Mar01, p.69]. It will be used in Section 5.3 below to relate Manin symbols to modular symbols. First, we will need some notation. For now, let R be a ring and K = Q(i). Define the following matrices in GL2 (OK ): Ã J=
i 0 0 1
!
à S=
0 i
!
à T =
1 0
1 1
!
0 1
à T0 =
1 0
!
1 1
Proposition 5.2.1. Consider the following homomorphism of left R[GL2 (OK )]-modules Ψ : R[GL2 (OK )] −→ R[P1 (K)] X X uM [M ] 7−→ uM ([M (∞)] − [M (0)]). M
M
The kernel of Ψ is equal to the left R[GL2 (OK )] ideal J = h[I] − [T ] − [T 0 ], [I] + [S], [I] − [J]i. Proof. First, we will show J ⊆ ker(Ψ). For this we simply evaluate Ψ on [I]−[T ]−[T 0 ],
62
on [I] + [S] and on [I] − [J]. We have Ψ([I] − [T ] − [T 0 ]) = [I(∞)] − [I(0)] − [T (∞)] + [T (0)] − [T 0 (∞)] + [T 0 (0)] = [∞] − [0] − [∞] + [1] − [1] + [0] =0 and Ψ([I] + [S]) = [I(∞)] − [I(0)] + [S(∞)] − [S(0)] = [∞] − [0] + [0] − [∞] =0 and Ψ([I] − [J]) = [I(∞)] − [I(0)] − [J(∞)] + [J(0)] = [∞] − [0] − [∞] + [0] = 0. The other direction, proving that ker(Ψ) ⊆ J , requires more work. Let W = P 1 M uM [M ] be a non-zero element of ker(Ψ). Let L(W ) ⊆ P (K) be the union P P of supports of M uM [M (∞)] and M uM [M (0)]. Furthermore, we define L(W ) = αmax (|α|2 + |β|2 ) β
and
∈L(W )
¯½ ¾¯ ¯ α ¯ 2 2 m(W ) = ¯¯ ∈ L(W ) : |α| + |β| = L(W ) ¯¯ β
where we assume (α, β) = 1. We will use elements of the ideal J to write down a W 0 congruent to W modulo J , but such that L(W 0 ) ≤ L(W ). Futhermore, if L(W 0 ) = L(W ) then we will have m(W 0 ) < m(W ). Iterating this process, we will see that W ∈ J . Let
α β
∈ L(W ) be such that |α|2 + |β|2 = L(W ). Let δ, γ be elements of OK
such that αγ − βδ = 1 and such that |γ| ≤ |β| and |δ| ≤ |α|. Then the matrices of
63
GL2 (OK ) satisfying M (∞) = α/β are of the form à M=
im α in (δ + kα)
!
im β in (γ + kβ)
for k ∈ OK and m, n ∈ {0, 1, 2, 3}. As L(W ) = |α|2 + |β|2 , we see that for such a matrix M to be in the support of W , we must have |δ + kα|2 + |γ + kβ|2 ≤ |α|2 + |β|2 . We will show first that for this to be true, we must have |k| < 2. First, suppose |k| ≥ 2. Then |δ + kα|2 + |γ + kβ|2 ≥ (|kα| − |δ|)2 + (|kβ| − |γ|)2 ≥ (2|α| − |δ|)2 + (2|β| − |γ|)2 = (|α| + (|α| − |δ|))2 + (|β| + (|β| − |γ|))2 ≥ |α|2 + |β|2 . Note, furthermore, that equality is only possible if |α| = |δ| and |β| = |γ|, but this contradicts the fact that αγ − βδ = 1. Thus we must have |k| < 2. Since k ∈ OK , the only possibilities are 1. k = 0; ∗ 2. k ∈ OK ; or
3. |k|2 = 2, i.e., k ∈ {1 + i, 1 − i, −1 + i, −1 − i}. Aside: We show why |α| = |δ| and |β| = |γ| implies that M cannot have deter∗ minant in OK . We have det(M ) = im+n (αγ − βδ). Under our assumption, we have ∗ |αγ| = |βδ|, so we need to show that we cannot have x − y ∈ OK if |x| = |y|. Suppose
this is the case and let x = a + bi and y = c + di, so that x − y = (a − c) + (b − d)i. Then either a = c and b = d ± 1 or b = d and a = c ± 1. Without loss of generality,
64
assume a = c and b = d ± 1. Then |x|2 = |y|2 a2 + b2 = c2 + d2
⇒
(d ± 1)2 = d2 ,
⇒ giving a contradiction. Note that the action of J is Ã
a b
!
à J=
c d
a b
!Ã
c d
i 0
!
à =
0 1
ia b
! ,
ic d
so, with (possibly repeated) applications of I − J, we may assume the matrix M has the form
à M=
α in (δ + kα)
! .
β in (γ + kβ)
Also, the action of S is given by Ã
a b
!
c d
à S=
a b
!Ã
c d
0 i
!
à =
1 0
b ia d ic
! ,
so, with (possibly repeated) applications of I − J and I + S, we may further assume the matrix M has the form à Mk =
α δ + kα
! .
β γ + kβ
Using the same techniques, we may assume that the only matrices N in the support of W such that N (0) = α/β are those of the form à Nj =
jα − δ α jβ − γ β
! ,
where |j|2 ∈ {0, 1, 2} and |jα − δ|2 + |jβ − γ|2 ≤ |α|2 + |β|2 . Each of the Nj 65
can be replaced, using S and J, by −M−j as follows. Note that I + JS ∈ J since I + JS = (I − J) + J(I + S). Then Nj − Nj (I + JS) = −Nj JS = −M−j . Let W 0 denote the element W ∈ ker(Ψ) with the above modifications. So now all matrices M in the support of W 0 with M (∞) = α/β are of the form à Mk =
α δ + kα β γ + kβ
! ,
with |δ + kα|2 + |γ + kβ|2 ≤ |α|2 + |β|2 . Furthermore, we have no matrices N in the support of W 0 with N (0) = α/β. The strategy from here is as follows. We will first consider the case where |k|2 = 2, and we will replace such matrices M by two matrices M 0 and M 00 , where M 0 will have the same form as M above but with |k| = 1 and M 00 will be such that M 00 (∞) 6= α/β 6= M 00 (0). These matrices will also satisfy m(M ) = m(M 0 − M 00 ), so that the number of α/β ∈ L(W ) will not increase. The next step is to work with the case |k| = 1. We will replace each M of this sort again with two matrices M 0 and M 00 . One of these will be in the form of Mk but with k = 0. The other will again be such that M 00 (∞) 6= α/β 6= M 00 (0). As in the |k|2 = 2 case, these matrices will satisfy m(M ) = m(M 0 − M 00 ). After these steps, the only matrix M left in W 0 with M (∞) = α/β is M0 , and there are no matrices left in W 0 with M (0) = α/β. Since W 0 ∈ ker(Ψ), we must have the coefficient uM0 of M0 in W 0 equal to 0. We thus decrease m(W 0 ) by at least one.
66
To aid in our analysis, we write α = a1 + a2 i β = b1 + b2 i δ = c1 + c2 i γ = d1 + d2 i. To deal with the case |k|2 = 2, we start by proving the following claim. Claim 5.2.2. Suppose k ∈ OK is such that |k|2 = 2, i.e., k ∈ {1 + i, 1 − i, −1 + i, −1 − i} and write k = k1 + k2 i. If |δ + kα|2 + |γ + kβ|2 ≤ |α|2 + |β|2 , then either |δ + k1 α|2 + |γ + k1 β|2 or |δ + k2 iα|2 + |γ + k2 iβ|2 is strictly less than |α|2 + |β|2 . Proof. Consider k = 1 + i. Our assumption then becomes |c1 + c2 i + (1 + i)(a1 + a2 i)|2 + |d1 + d2 i + (1 + i)(b1 + b2 i)|2 = |(a1 − a2 + c1 ) + (a1 + a2 + c2 )i|2 + |(b1 − b2 + d1 ) + (b1 + b2 + d2 )i|2 = 2(a21 + a22 ) + 2a1 c1 − 2a2 c1 + 2a1 c2 + 2a2 c2 + c21 + c22 + 2(b21 + b22 ) + 2b1 d1 − 2b2 d1 + 2b1 d2 + 2b2 d2 + d21 + d22 ≤ a21 + a22 + b21 + b22 , i.e., a21 + a22 + 2a1 c1 − 2a2 c1 + 2a1 c2 + 2a2 c2 + c21 + c22 + b21 + b22 + 2b1 d1 − 2b2 d1 + 2b1 d2 + 2b2 d2 + d21 + d22 ≤ 0.
67
(5.1)
Suppose, by way of contradiction, that the claim is false for k = 1 + i. Then we have |α + δ|2 + |β + γ|2 = a21 + a22 + 2a1 c1 + 2a2 c2 + c21 + c22 + b21 + b22 + 2b1 d1 + 2b2 d2 + d21 + d22 ≥ a21 + a22 + b21 + b22 , that is, 2a1 c1 + 2a2 c2 + c21 + c22 + 2b1 d1 + 2b2 d2 + d21 + d22 ≥ 0.
(5.2)
We also have |iα + δ|2 + |iβ + γ|2 = a21 + a22 − 2a2 c1 + 2a1 c2 + c21 + c22 + b21 + b22 − 2b2 d1 + 2b1 d2 + d21 + d22 ≥ a21 + a22 + b21 + b22 , that is, −2a2 c1 + 2a1 c2 + c21 + c22 − 2b2 d1 + 2b1 d2 + d21 + d22 ≥ 0.
(5.3)
Adding together 5.2 and 5.3, we see that 2a1 c1 + 2a2 c2 + c21 + c22 + 2b1 d1 + 2b2 d2 + d21 + d22 − 2a2 c1 + 2a1 c2 + c21 + c22 − 2b2 d1 + 2b1 d2 + d21 + d22 ≥ 0, but, as |δ|2 + |γ|2 ≤ |α|2 + |β|2 , the left hand side of 5.4 is less than or equal to a21 + a22 + b21 + b22 + 2a1 c1 + 2a2 c2 + 2b1 d1 + 2b2 d2 − 2a2 c1 + 2a1 c2 + c21 + c22 − 2b2 d1 + 2b1 d2 + d21 + d22 , i.e., a21 + a22 + b21 + b22 + 2a1 c1 + 2a2 c2 + 2b1 d1 + 2b2 d2 − 2a2 c1 + 2a1 c2 + c21 + c22 − 2b2 d1 + 2b1 d2 + d21 + d22 ≥ 0.
68
(5.4)
If we have “strictly greater than” in the above, we get a contradiction with (5.1). Equality requires that |δ|2 + |γ|2 = |α|2 + |β|2 . Since |δ|2 ≤ |α|2 and |γ|2 ≤ |β|2 , equality can only happen if |δ|2 = |α|2 and |γ|2 = |β|2 , which contradicts the fact that αγ − βδ = 1. Thus we must have either |α + δ|2 + |β + γ|2 or |iα + δ|2 + |iβ + γ|2 is less than |α|2 + |β|2 . The other cases are treated similarly. For the first step, i.e. |k|2 = 2, we can apply Claim 5.2.2 above as follows: Let t1 ∈ {k1 , k2 i} be such that |δ + t1 α|2 + |γ + t1 β| < |α|2 + |β|2 and let t2 be the other element of {k1 , k2 i}. Then Ã
t2 α δ + t1 α t2 β γ + t1 β
!
à T =
t2 α δ + t1 α
!Ã
t2 β γ + t1 β
1 1
!
à =
0 1
t2 α δ + kα
!
t2 β γ + kβ
and Ã
t2 α δ + t1 α t2 β γ + t1 β
!
à T0 =
t2 α δ + t1 α
!Ã
t2 β γ + t1 β
1 0
!
à =
1 1
δ + kα δ + t1 α γ + kβ γ + t1 β
Since t2 = im for some integer m, we can apply I − J to M until we have à M=
Defining
and
à M 00 =
! .
t2 β γ + kβ Ã
M0 =
t2 α δ + kα
t2 α δ + t1 α
!
t2 β γ + t1 β δ + kα δ + t1 α γ + kβ γ + t1 β
69
! ,
! .
we then have à M 0 (I − T − T 0 ) = à =
t2 α δ + t1 α t2 β γ + t1 β t2 α δ + t1 α
! (I − T − T 0 ) !
à −
t2 β γ + t1 β
t2 α δ + kα t2 β γ + kβ
!
à −
δ + kα δ + t1 α
!
γ + kβ γ + t1 β
= M 0 − M − M 00 , so, modulo J , we may replace M by M 0 − M 00 . Note that M 0 can be replaced by à Mk0 =
α δ + k0α
! ,
β γ + k0β
with |k 0 |2 = 1, a case we will treat shortly. Also, we have L(M 00 ) ≤ L(M ) and M 00 (∞) 6= α/β 6= M 00 (0). Recall that when L(W 0 ) = L(W ), we need m(W 0 ) < m(W ). In this step, we are replacing one matrix M with two, M 0 and M 00 , but since |δ + t1 α|2 + |γ + t1 β| < |α|2 + |β|2 , we have m(M ) = m(M 0 − M 00 ). (Note that M 00 (∞) = (δ + kα)/(γ + kβ) was already in L(W 0 ) as M 00 (∞) = M (0).) We are not decreasing m(W ) in this step, but we are not increasing it either. In later steps we will obtain a decrease in either L(W 0 ) or m(W 0 ). So now the only matrices M remaining in the support of W with M (∞) = α/β are those of the form
à Mk =
α δ + kα β γ + kβ
! ,
(5.5)
where either k = 0 or |k|2 = 1 and |δ + kα|2 + |γ + kβ|2 ≤ |α|2 + |β|2 (by assumption for those starting out with |k|2 = 1 and by design for those coming initially from matrices with |k|2 = 2). Now we will use elements of J to eliminate the Mk for |k| = 1. In particular, we will use I − T − T 0 . We can first use repeated applications of I − J to replace Mk by
70
˜ k with left hand column given by the transpose of (−kα −kβ), i.e., M Ã ˜k = M Defining
à 0
M =
−kα δ −kβ γ
−kα δ + kα
!
−kβ γ + kβ
!
à 00
and M =
.
δ δ + kα
!
γ γ + kβ
,
we then have à ˜ k (I − T − T 0 ) = M à =
−kα δ + kα −kβ γ + kβ −kα δ + kα
! (I − T − T 0 ) !
à −
−kβ γ + kβ
−kα δ −kβ γ
!
à −
δ δ + kα
!
γ γ + kβ
˜ k − M 0 − M 00 . =M ˜ k by M 0 + M 00 , where M 0 can be replaced by M0 and M 00 Thus we can replace M will satisfy: 1. M 00 (∞) 6=
α β
6= M 00 (0) and
2. L(M 00 ) ≤ L(W ). ˜ k ) = m(M 0 + M 00 ). (Note that M 00 (0) = (δ + kα)/(γ + kβ) Furthermore we have m(M ˜ k (0).) was already in L(W 0 ) as M 00 (0) = M Finally the only matrix in the support of W 0 with either M (∞) or M (0) equal to α/β is M0 . Thus we must have uM0 = 0 and so m(W 0 ) is decreased by at least one.
5.3
Manin Symbols
Recall that in Section 5.1, we showed that we wish to compute H0 (Γ, St ⊗ M ) = (S0 ⊗ M )Γ /∂((S1 ⊗ M )Γ ), 71
where ∂ : (S1 ⊗ M )Γ −→ (S0 ⊗ M )Γ is the boundary map defined by ∂([v1 , v2 , v3 ]) = [v1 , v3 ] − [v1 , v2 ] − [v2 , v3 ]. If M = R, we are computing the set of formal R-linear sums of symbols [v] = [v1 , v2 ], where each vi is a unimodular column in O2 modulo the R[Γ]-module generated by the following elements: 1. [v2 , v1 ] + [v1 , v2 ]; 2. [v] if det(v) = 0; 3. [v1 , v3 ] − [v1 , v2 ] − [v2 , v3 ]; and 4. [v] − γ[v] for all γ ∈ Γ, where the vi run over all unimodular columns in O2 . We call the [v] = [v1 , v2 ] modular symbols. Note that if we compute the same thing but without the [v] − γ[v] relations, then we are just computing the Steinberg module St. We will follow the approach of Wiese [Wie05, p.25-29] in using Proposition 5.2.1 from Section 5.2 to relate Manin symbols to modular symbols. Manin symbols provide us with an explicit, computationally friendly description of the homology we wish to compute. Proposition 5.3.1 below gives us the first step in the transition from modular symbols to Manin symbols. We define another matrix: Ã L :=
We will also use
1 Ã
L2 =
1 −1
! .
0
0 −1 1 −1
! .
In the following we will use the notation α/β for the unimodular column [α β]T . In particular, we will use 0 to denote [0 1]T and ∞ to denote [1 0]T . 72
Proposition 5.3.1. The following homomorphism of R-modules is an isomorphism: Φ : R[GL2 (OK )]/I −→ St M 7−→ [M (0), M (∞)] where I = R[GL2 (OK )](I − J) + R[GL2 (OK )](I + S) + R[GL2 (OK )](I + L + L2 ). Proof. To prove that Φ is surjective, first note that, in the Steinberg module St, we have [v1 , v2 ] = [v1 , 0] + [0, v2 ] = −[0, v1 ] + [0, v2 ], so it suffices to show [0, v 0 ] is in the image of Φ, where v 0 is any unimodular column in O2 . To see this, we use continued fractions to write down a sum of matrices in SL2 (OK ) which, under Φ, will be mapped to [0, v 0 ]. For this algorithm, we rely on the fact that our fields K are Euclidean. We use continued fractions to write a given modular symbol of the form [0, α] as a finite sum of symbols of the form [γ(0), γ(∞)] with γ ∈ SL2 (OK ). We set r0 = α and rn = 1/(rn−1 − an−1 ) and define an = floor(rn ). Here we define the floor function on Q(i) as follows: floor(r + si) for r, s ∈ Q is equal to a + bi with a and b the least integers such that |a − r| ≤ 1/2 and |b − s| ≤ 1/2. We then define the convergents of the continued fractions as follows: p−2 = 0
q−2 = 1
p−1 = 1
q−1 = 0
pn = an pn−1 + pn−2 so that
qn = an qn−1 + qn−2
1 pn = a0 + qn a1 + a2 + 1
1 ···+ a1 n
73
and α=
pk . qk
As in continued fractions for Z, we have pn qn−1 − pn−1 qn = (−1)n+1 . We can now rewrite the modular symbol [0, α] as ¸ k · k k X X X pn−1 pn [0, α] = , = [γn (0), γn (∞)] = Φ(γn ), q q n−1 n n=−1 n=−1 n=−1 where
à γn =
(−1)n+1 pn pn−1 (−1)n+1 qn qn−1
! .
We now show that the kernel of Φ is equal to I = R[GL2 (OK )](I − J) + R[GL2 (OK )](I + S) + R[GL2 (OK )](I + L + L2 ). We start by showing that ker(Φ) = ker(Ψ) where Ψ is the map in Proposition 5.2.1. Define another map π by π : St −→ R[P1 (K)] [v1 , v2 ] 7−→ v2 − v1 . Then we have Ψ = π ◦ Φ, so certainly ker(Φ) ⊆ ker(Ψ). To show the other inclusion, P suppose M uM M ∈ ker(Ψ), i.e., Ψ
à X M
! uM M
=
X
uM M (0) −
M
X M
74
uM M (∞) = 0.
Applying Φ instead of Ψ to this element of ker(Ψ), we then have Φ
à X
! uM M
=
M
X
uM [M (0), M (∞)]
M
=
X
uM [M (0), ∞] +
M
=
X
X
uM [∞, M (∞)]
M
uM [M (0), ∞] −
M
X
uM [M (∞), ∞]
M
= 0, and so ker(Ψ) ⊆ ker(Φ). Finally, we need to show that ker(Φ) can be written in the form claimed, i.e., that ker(Φ) = R[GL2 (OK )](I − J) + R[GL2 (OK )](I + S) + R[GL2 (OK )](I + L + L2 ). Currently, we have that ker(Φ) = R[GL2 (OK )](I − J) + R[GL2 (OK )](I + S) + R[GL2 (OK )](I − T − T 0 ). First, note that in R[GL2 (OK )](I − J) + R[GL2 (OK )](I + S) we have both I + S˜ and I − J˜ where à S˜ = JS =
0 −1 1
0
!
à and J˜ = J 2 =
as I + S˜ = (I − J) + J(I + S) and I − J˜ = (I − J) + J(I − J).
75
−1 0 0
1
!
We then also have I + J˜S˜J˜ as ˜ + J(I ˜ + S) ˜ − J˜S(I ˜ − J). ˜ I + J˜S˜J˜ = (I − J) Now, to see that the two forms of the kernel are the same, we write ˜ + T 0 (I + S) ˜ = I + T S˜ + T 0 S˜ (I − T − T 0 ) + T (I + S) = I + L + L2 and
˜ = I − LJ˜S˜J˜ − L2 J˜S˜J˜ (I + L + L2 ) − (L + L2 )(I + J˜S˜J) = I − T − T 0.
Our discussion thus far is only sufficient for computing weight two modular forms. We now extend this so that we can compute higher weight forms and we also incorporate the Γ relations [v] − γ[v]. ¯ ` and let M = V be a Serre weight with left R[GL2 (OK )] action, Let R = F as defined in Section 3.1. We consider the module Γ (R[GL2 (OK )] ⊗R M ), where Γ acts diagonally on the left and we have the natural right R[GL2 (OK )] action, i.e., (h ⊗ v)g = (hg ⊗ v). In the following theorem and subsequent proposition we use Proposition 5.3.1 to write the homology group we wish to compute in terms of Manin symbols. Theorem 5.3.2. Let N denote the R-module Γ (R[GL2 (OK )] ⊗R M ) as described above. Then the following sequence of R-modules is exact: 0 → N (I − J) + N (I + S) + N (I + L + L2 ) → N → H0 (Γ, St ⊗ M ) → 0. Proof. Proposition 5.3.1 gives the exact sequence 0 → I → R[GL2 (OK )] → St → 0
76
where I = R[GL2 (OK )](I − J) + R[GL2 (OK )](I + S) + R[GL2 (OK )](I + L + L2 ). Since M is a free R-module, the following sequence of R[Γ]-modules is also exact: 0 → N 0 (I − J) + N 0 (I + S) + N 0 (I + L + L2 ) → N 0 → St ⊗ M → 0. We then need only take Γ-coinvariants to achieve the desired exact sequence. We need one more step to get to the module we will actually be computing. This is the content of the following proposition. Proposition 5.3.3. Let X denote the R-module R[Γ\GL2 (OK )] ⊗R M with right GL2 (OK )-action given by (Γh⊗v)g = Γhg ⊗g −1 v. Then there is a right R[GL2 (OK )]module isomorphism between N =Γ (R[GL2 (OK )] ⊗R M ) and X. Proof. The isomorphism is simply given by g ⊗ v 7→ g ⊗ g −1 v. The R-module X = R[Γ\GL2 (OK )] ⊗R M is the module of Manin symbols and is the basic module we will use in computations. For Γ = Γ0 (n), the coset representatives of Γ\GL2 (OK ) are in one-to-one correspondence with P1 (n), the projective line over OK /n. We use the notation (c : d) with c, d ∈ OK to denote such a coset representative.
5.4
Γ1 (n) and Characters
The treatment in Section 5.3 is sufficient for dealing with Γ = Γ0 (n). We will often want to compute forms for Γ1 (n), both for theoretical and computational reasons. The above method can also be used for the Γ1 (n) case (Figueiredo, in [Fig99], does in fact use this method). However, we will follow the approach used by Wiese [Wie05] for the Γ1 (n) case, as it is computationally advantageous to do so. In this section, we describe this method.
77
In the Γ1 (n) case, we will use a character ∼ ¯ ∗. ε : Γ0 (n) ³ Γ1 (n)\Γ0 (n) → (OK /n)∗ → F `
Remarks 5.4.1. 1. When the modular form in question is associated to a Galois representation ρ, this character will correspond to the character ερ of that representation. 2. We recover the Γ0 (n) case by taking Γ1 (n) with the trivial character. We define a slight variation on the weight module M , which takes into account the action of the character ε. This we define as ¯ ε, M ε = M ⊗F¯` F ` ¯ ε denotes a copy of F ¯ ` with action of Γ0 (n) by ε−1 . In particular, for imaginary where F ` quadratic fields K we have ¯ ε. M ε = detaτ ⊗ detaτ 0 ⊗ Symbτ −1 ⊗ Symbτ 0 −1 ⊗ F ` When ` splits in K, the embeddings τ and τ 0 will be the embeddings of kp and k¯p in ¯ ` . When ` is inert in K, the embedding τ 0 will be τ ◦ Frob` . F Computing cohomological mod ` modular forms for Γ1 (n) with character ε then amounts to computing simultaneous eigenvectors for the Hecke operators on ¡ Γ1 (n)\Γ0 (n) Γ1 (n)
¢ (R[GL2 (OK )] ⊗ M ε ) ,
modulo the relations used in Proposition 5.3.1. Here Γ1 (n)\Γ0 (n) is acting diagonally ¡ ¢ ¯ε . on the left on Γ (n) R[GL2 (OK )] ⊗ M ⊗ F 1
`
Proposition 5.4.2. Consider the R-module ¯ ε, X = R[Γ0 (n)\GL2 (OK )] ⊗ M ⊗ F ` 78
where GL2 (OK ) acts on the right by (h ⊗ v ⊗ r)g = (hg ⊗ g −1 v ⊗ r) and Γ1 (n)\Γ0 (n) acts on the left by g(h ⊗ v ⊗ r) = (gh ⊗ v ⊗ ε(g)r). To compute cohomological mod ` modular forms for Γ1 (n) with character ε of the form described above, we can equivalently compute simultaneous eigenvectors for the Hecke operators on the space X modulo the relations used in Proposition 5.3.1. Proof. First we apply the isomorphism ¡ Γ1 (n)\Γ0 (n) Γ1 (n)
¢ (R[GL2 (OK )] ⊗ M ε ) ∼ =Γ0 (n) (R[GL2 (OK )] ⊗ M ε ) .
Then we apply the isomorphism of Proposition 5.3.3.
5.5
Hecke Operators
In this section, we define Hecke operators on the space of modular symbols H0 (Γ, St⊗ ¯ ε ), with Γ = Γ1 (n), but also with left coinvariants by Γ1 (n)\Γ0 (n) via the M ⊗F `
character ε, as described in Section 5.4. (To compute Hecke operators, we convert Manin symbols to modular symbols, compute the action of the Hecke operators there, and then convert back to Manin symbols. We use the results of Section 5.3 to convert back and forth.) Let p be a prime ideal of OK which is relatively prime to the level n and let π be a generator for p. We define a set ∆p ⊂ GL2 (K) by (Ã ∆p =
a b c d
!
à ∈ M2 (OK ) : ad − bc = π,
a b c d
!
à ≡
u ∗ 0 π
!
) mod n ,
∗ where u ∈ OK . Using the fact that K is Euclidean, one can easily show that
Γ1 (n)∆p = ∆p Γ1 (n) = ∆p
79
and ∆p can be written as a disjoint union as
∆p =
[
à Γ1 (n) · σa
a,x
à where σa ≡
1/a 0
a
x
0 π/a
! ,
!
mod n, a ∈ {1, π}, and x runs over representatives of 0 a OK /(π/a). Furthermore, since we take coinvariants via the character action on Γ1 (n)\Γ0 (n) and σa ∈ Γ0 (n), we may define the Hecke operator Tp by à Tp ([v1 , v2 ] ⊗ P ⊗ Q) =ε(π)
+ x
π 0
!
([v1 , v2 ] ⊗ P ⊗ Q) 0 1 Ã ! X 1 x ([v1 , v2 ] ⊗ P ⊗ Q) . 0 π mod π
The Hecke operator Tp is well-defined since Γ1 (n)∆p = ∆p Γ1 (n). Also, the Hecke operator Tp is independent of the choice of generator π, since any other generator will ! Ã ² 0 ∗ be of the form ²π for ² ∈ OK ∈ Γ1 (n). and then T²π = JTπ where J = 0 1
80
Computational Evidence 6.1
Algorithm for Computing Modular Forms
To compute these higher weight mod ` modular forms over Q(i), I wrote a program in C using the PARI library [The05]. In this section, I give an outline of how the program works. Let Γ = Γ1 (n) for some ideal n ⊂ OK with character ∼ ¯ ∗. ε : Γ\Γ0 (n) → (OK /n)∗ → F `
Let V be a Serre weight for K = Q(i) and denote by V ε this same Serre weight but with character action, so we have ¯ ε. V ε = detaτ ⊗ detaτ 0 ⊗ Symbτ −1 ⊗ Symbτ 0 −1 ⊗ F ` The program computes cohomological
mod ` modular forms of level n, character ² ¯ ` . The program executes the and weight V . For notational convenience, set R = F following steps. 1. Compile a list of basic Manin symbols. The Manin symbols module, described in Section 5.3 is the R-module X = R[Γ0 (n)\GL2 (OK )] ⊗R V ε . To compute generators for this module, we compute a set of coset representatives (c : d) for Γ0 (n)\GL2 (OK ), which is in one-to-one correspondence with P1 (n), the projective line over OK /n (see, e.g., [Byg98, p 29]). Recall that each Symb can be viewed as the space of homogeneous polynomials of degree b in two variables with coefficients in R. We can take as a set of generators elements of
81
the form (c : d) ⊗ X m Y n ⊗ Z r W s , where m + n = bτ − 1 and r + s = bτ 0 − 1. In practice, we only store the coset representatives (c : d) and then use a “column number” to indicate the weight (ordered by the exponent of Y and then the exponent of W ). 2. Quotient out the above module by the two and three term relations described in Section 5.3, namely I − J, I + S and I + L + L2 . For this we create a 3m × m matrix, where m is the number of basic Manin symbols computed in Step 1. For each basic Manin symbol, we have three rows, one for each relation. We have to be careful with the character action here. For each row, after computing the action of one of the relations on a basic Manin symbol, we have to write the result again in terms of the basic Manin symbols (for the particular coset representatives we chose for R[Γ0 (n)\GL2 (OK )]). Here we use the left Γ1 (n)\Γ0 (n) action described in Proposition 5.4.2. For example, if after computing the action of one of these relations, we have some (c0 : d0 ) ⊗ X m Y n ⊗ Z r W s in the result, with (c0 : d0 ) = h(c : d) for h ∈ Γ1 (n)\Γ0 (n) and (c : d) ⊗ X m Y n ⊗ Z r W s one of our chosen coset representatives, then we write (c0 : d0 ) ⊗ X m Y n ⊗ Z r W s = ε−1 (h)((c : d) ⊗ X m Y n ⊗ Z r W s ). We then row reduce this matrix to find generators for the space H0 (Γ, St ⊗ V ), as well as linear combinations of these generators for each of the basic Manin symbols. This matrix can be quite large and the reduction of it presents the biggest bottleneck for the program in terms of speed and memory. 3. Once we have the generators for the space and the linear combinations for the basic Manin symbols, we are ready to compute Hecke operators. We compute each Hecke operator Tp as a matrix acting on the generators of the space. Our method of computing Hecke operators is as follows. Take each generator, 82
of the form (c : d) ⊗ X m Y n ⊗ Z r W s and convert it to a modular symbol, i.e., an element of the module ¯ ε ). H0 (Γ, St ⊗ M ⊗ F ` Ã We convert it by first lifting the coset representative (c : d) to a matrix
a b
!
c d GL2 (OK ). (Actually, we can take the lift in SL2 (OK ).) We then convert this to a modular symbol by the following map ÃÃ
a b
!
! ⊗ X mY n ⊗ Z r W s
7→ c d µ· ¸ ¶ b a m n r s ¯ ) (−¯ , ⊗ (bX − dY ) (−cX + aY ) ⊗ (¯bZ − dW cZ + a ¯W ) d c Ã
a b
!
has determinant 1, we do not need to c d account for the action via the deta components of the weight module V .
Note that, since the matrix
We then compute the left action of the Hecke operator Tp on this modular symbol, resulting in a sum of modular symbols. Using basic relations of the modular symbols, we write each as a sum of the form: µ·
¸ ¶ b a , ⊗ P (X, Y ) ⊗ Q(Z, W ) d c ¶ ³h a i ´ µ· b ¸ = 0, ⊗ P (X, Y ) ⊗ Q(Z, W ) − 0, ⊗ P (X, Y ) ⊗ Q(Z, W ) c d
We then use the continued fractions method to convert each of these terms to a sum of basic Manin symbols which in turn can be written in terms of the generators. When converting from modular symbols back to Manin symbols, we again have to be careful with the weight action. The result gives one row of the Hecke operator Tp . We repeat this process for 83
∈
each generator to get the full matrix for Tp .
6.2
Torsion
Early in section 5.1 we assume that the torsion in Γ is invertible in the commutative ¯ ` , though we ring R. In our case, Γ is a congruence subgroup of GL2 (OK ) and R = F may think of R as Fq where q is some power of `. This assumption about torsion is not a strong assumption, since we can make sure that, as long as the level n is large enough, the congruence subgroup Γ will be torsion free. In our examples, we have K = Q(i) and Γ = Γ1 (n) for some level n. Now suppose A ∈ GL2 (K) \ K ∗ is a torsion element of prime power, i.e., there is some prime p such that Ap = I. Now consider K[A]. We have K[A] ∼ = K[x]/(x2 − T x + D), where T is the trace of A and D is the determinant of A. Since A 6∈ K ∗ , we may assume x2 − T x + D is irreducible and so K[A] is a field, and a quadratic extension of K. Since A is a pth root of unity, we also have K[A] ∼ = K[ζp ]. Then we have a quadratic extension of K of the form K[ζp ]. Since K = Q(i), this implies that p is 2 or 3. We will not use ` = 2 for any of our examples (since 2 is ramified in K and hence not covered by the BDJ conjecture), but we do have some examples with ` = 3, so consider p = 3. Then in the polynomial above we have T = −1 and D = 1. The matrix A is in the congruence subgroup Γ1 (n), so we have à A=
a b c d
!
à ≡
∗ ∗ 0 1
! mod n.
Since the determinant D = 1, this implies that a ≡ 1 mod n. Then T = −1 implies 2 ≡ −1 mod n, which in turn implies that n | 3OK . None of our examples have n | 3OK , and so all of our examples satisfy the condition on torsion in Γ.
84
6.3
Modular Forms Corresponding to Elliptic Curves
In Section 4.1, we determined which elliptic curves over Q(i) of those in [Cre84] were supersingular and had good reduction at ` = 7. For the representations corresponding to these elliptic curves we know the level (from the conductor of the elliptic curve) and we know that the character will be trivial. We computed the set of predicted weights using the BDJ conjecture. In this section we give tables for each of the levels with, for some small primes p, the values ap (E) associated to the elliptic curve and the systems of eigenvalues ap (f ) which we found at the predicted levels and weights. Recall that the values ap (E) associated to the elliptic curve are given by ap = 1 + N m(p) − Mp , where N m(p) is the norm of p and Mp is the number of points on the curve E modulo p (including the point at infinity). For each of the four levels tested here, we found the corresponding systems of eigenvalues in all of the predicted weights.
85
Table 6.1: Elliptic curve, conductor n = 6 + 6i, considered mod 7
p ap(E) ap(f ) 1 − 2i −2 5 1 + 2i −2 5 2 − 3i −2 5 2 + 3i −2 5 1 − 4i 2 2 1 + 4i 2 2 2 − 5i 6 6 2 + 5i 6 6 1 − 6i 6 6 1 + 6i 6 6 4 − 5i −6 1 4 + 5i −6 1 2 − 7i −2 5 2 + 7i −2 5 5 − 6i −2 5 5 + 6i −2 5
p ap(E) ap(f ) 3 − 8i 10 3 3 + 8i 10 3 5 − 8i −6 1 5 + 8i −6 1 4 − 9i 2 2 4 + 9i 2 2 1 − 10i −18 3 1 + 10i −18 3 3 − 10i −2 5 3 + 10i −2 5 7 − 8i 18 4 7 + 8i 18 4 11 −6 1 4 − 11i −6 1 4 + 11i −6 1
86
Table 6.2: Elliptic curve, conductor n = 9 + 7i, considered mod 7
p ap(E) ap(f ) 1 + 2i 0 0 3 4 4 2 − 3i −4 3 1 − 4i 0 0 1 + 4i −6 1 2 − 5i −6 1 2 + 5i 0 0 1 − 6i 2 2 1 + 6i 2 2 4 − 5i −6 1 4 + 5i 6 6 2 − 7i 6 6 2 + 7i −6 1 5 − 6i −10 4 5 + 6i 8 1
p ap(E) ap(f ) 3 − 8i 2 2 3 + 8i 2 2 5 − 8i 12 5 5 + 8i −6 1 4 − 9i 8 1 4 + 9i 8 1 1 − 10i −6 1 1 + 10i −18 3 3 − 10i 2 2 3 + 10i 2 2 7 − 8i 6 6 7 + 8i 18 4 11 −4 3 4 − 11i 6 6 4 + 11i −6 1
87
Table 6.3: Elliptic curve, conductor n = 15, considered mod 7
p ap(E) ap(f ) 1+i −1 6 2 − 3i −2 5 2 + 3i −2 5 1 − 4i 2 2 1 + 4i 2 2 2 − 5i −2 5 2 + 5i −2 5 1 − 6i −10 4 1 + 6i −10 4 4 − 5i 10 3 4 + 5i 10 3 2 − 7i −10 4 2 + 7i −10 4 5 − 6i −2 5 5 + 6i −2 5
p ap(E) ap(f ) 3 − 8i 10 3 3 + 8i 10 3 5 − 8i −6 1 5 + 8i −6 1 4 − 9i 2 2 4 + 9i 2 2 1 − 10i 6 6 1 + 10i 6 6 3 − 10i 14 0 3 + 10i 14 0 7 − 8i 2 2 7 + 8i 2 2 11 −6 1 4 − 11i −6 1 4 + 11i −6 1
88
Table 6.4: Elliptic curve, conductor n = 19 + 9i, considered mod 7
p ap(E) ap(f ) 1 − 2i 3 3 1 + 2i 0 0 3 −5 2 2 − 3i 5 5 1 + 4i 3 3 2 − 5i −6 1 2 + 5i 0 0 1 − 6i 2 2 1 + 6i −7 0 4 − 5i 3 3 4 + 5i −3 4 2 − 7i −12 2 2 + 7i 3 3 5 − 6i −1 6 5 + 6i −1 6
p ap(E) ap(f ) 3 − 8i −16 5 3 + 8i 2 2 5 − 8i 12 5 5 + 8i −15 6 4 − 9i −10 4 4 + 9i 8 1 1 − 10i −15 6 1 + 10i 0 0 3 − 10i 2 2 3 + 10i 11 4 7 − 8i −3 4 7 + 8i 0 0 11 −4 3 4 − 11i 6 6 4 + 11i 3 3
89
6.4
Modular Forms Corresponding to Representations from Polynomials
6.4.1 Dihedral Group D4 In Section 4.2.2 we computed a
mod 5 representation ρ with level n = 29 and
quadratic character ε29 , which factors through a Galois extension L over K = Q(i) with G = Gal(L/K) ∼ = D4 . In the following table we recall the correspondence between orders of elements of G and the traces of the images of those elements under ρ. This representation ρ is a base change from a representation over Q, so we give the traces for both the case where the prime p of K splits over Q and for the case where it is inert over Q. order
trace (split)
trace (inert)
1
2
2
2
0, 3
2
4
0
3
Note that in the split case there is some ambiguity in the trace for order 2 elements, depending on whether the element is central. For each of those, I computed the restriction of the representation to the decomposition group at that prime to determine whether it should be 0 or 3. In the following table we list, for some small primes p, the order of Frobp as discussed above along with the coefficients ap of the corresponding system of eigenvalues found. For this example, we found the modular form for two of the four predicted weights. Further investigation is required to understand why the form did not show up as predicted in two of the weights. Table 4.3 in Section 4.2.2 lists the predicted weights along with the level, character and ` for each modular form predicted by the conjecture to correspond to ρ.
90
Table 6.5: Galois group D4 , level n = 29, considered mod 5
p o(Frobp) 1+i 4 3 4 2 − 3i 2 2 + 3i 2 1 − 4i 4 1 + 4i 4 1 − 6i 4 1 + 6i 4 4 − 5i 2 4 + 5i 2 7 2 2 − 7i 2 2 + 7i 2 5 − 6i 2 5 + 6i 2 3 − 8i 4 3 + 8i 4
ap 0 3 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0
p o(Frobp) 5 − 8i 2 5 + 8i 2 4 − 9i 4 4 + 9i 4 1 − 10i 2 1 + 10i 2 3 − 10i 1 3 + 10i 1 7 − 8i 4 7 + 8i 4 11 2 4 − 11i 4 4 + 11i 4 7 − 10i 1 7 + 10i 1
91
ap 0 0 0 0 0 0 2 2 0 0 2 0 0 2 2
6.4.2 Alternating Group A4 In Section 4.2.3 we computed two
mod 3 representations, one of level n = 61 and
the other of level n = 79. Both have trivial character and factor through a Galois extension M over K = Q(i) with G = Gal(M/K) ∼ = Aˆ4 . In the following table we recall the correspondence between orders of elements of G and the traces of the images of those elements under ρ. This representation ρ is a base change from a representation over Q, so we give the traces for both the case where the prime p of K splits over Q and for the case where it is inert over Q. order
trace (split)
trace (inert)
1
2
2
2
1
2
3
2
2
4
0
1
6
1
2
In the following tables we list, for some small primes p, the order of Frobp as discussed above along with the coefficients ap of the corresponding system of eigenvalues found. For both examples, we found the modular form for all the predicted weights. Tables 4.4 and 4.5 in Section 4.2.3 list the predicted weights along with the level, character and ` for each modular form predicted by the conjecture to correspond to ρ.
92
Table 6.6: Galois group A4 , level n = 61, considered mod 3
p o(Frobp) 1+i 4 1 − 2i 6 1 + 2i 6 2 − 3i 6 2 + 3i 6 1 − 4i 6 1 + 4i 6 2 − 5i 3 2 + 5i 3 1 − 6i 4 1 + 6i 4 4 − 5i 4 4 + 5i 4 7 3 2 − 7i 4 2 + 7i 4
ap 0 1 1 1 1 1 1 2 2 0 0 0 0 2 0 0
p o(Frobp) 3 − 8i 3 3 + 8i 3 5 − 8i 4 5 + 8i 4 4 − 9i 3 4 + 9i 3 1 − 10i 6 1 + 10i 6 3 − 10i 6 3 + 10i 6 7 − 8i 4 7 + 8i 4 11 4 4 − 11i 6 4 + 11i 6 7 − 10i 4 7 + 10i 4
93
ap 2 2 0 0 2 2 1 1 1 1 0 0 1 1 1 0 0
Table 6.7: Galois group A4 , level n = 79, considered mod 3
p o(Frobp) 1+i 4 1 − 2i 4 1 + 2i 4 2 − 3i 3 2 + 3i 3 1 − 4i 4 1 + 4i 4 2 − 5i 6 2 + 5i 6 1 − 6i 6 1 + 6i 6 4 − 5i 3 4 + 5i 3 7 3 2 − 7i 6 2 + 7i 6 5 − 6i 3 5 + 6i 3
ap 0 0 0 2 2 0 0 1 1 1 1 2 2 2 1 1 2 2
p o(Frobp) 3 − 8i 3 3 + 8i 3 5 − 8i 2 5 + 8i 2 4 − 9i 6 4 + 9i 6 1 − 10i 3 1 + 10i 3 3 − 10i 6 3 + 10i 6 7 − 8i 6 7 + 8i 6 11 3 4 − 11i 6 4 + 11i 6 7 − 10i 4 7 + 10i 4
94
ap 2 2 1 1 1 1 2 2 1 1 1 1 2 1 1 0 0
6.5
Modular Forms Corresponding to Representations from CFT
6.5.1 Dihedral Group D3 For these D3 examples over K = Q(i), we compute the values tr(ρ(Frobp )) for each prime p by computing the product of 1. the inertia degree of p ⊂ OK in the quadratic extension L, and 2. the order of a prime P ⊂ OL above p in the class group of L. This product gives us the order of Frobp in the Galois group, which is isomorphic to ¯ ` ), D3 . We denote this order by o(Frobp ). Recall that, from the image of ρ in GL2 (F we have the following correspondence between orders of elements and the traces of the images of the elements under ρ: order
trace
1
2
3
−1
2
0
In the following tables we list, for some small primes p, the order of Frobp as discussed above along with the coefficients ap of the systems of eigenvalues (considered mod 5 and
mod 7). There are three tables, one for each of the levels n = 8 + 17i,
n = 13 + 28i and n = 8 + 35i. In all cases, we found the corresponding systems of eigenvalues in all weights predicted for both ` = 5 and for ` = 7. Table 4.6 in Section 4.3.1 lists the predicted weights along with the level, character and ` for each modular form predicted by the conjecture to correspond to ρ.
95
Table 6.8: Galois group D3 , level n = 8 + 17i, considered mod 5 and mod 7
p o(Frobp) 1+i 3 1 − 2i 2 1 + 2i 3 3 2 2 − 3i 3 2 + 3i 2 1 − 4i 3 1 + 4i 3 2 − 5i 2 2 + 5i 2 1 − 6i 3 1 + 6i 2 4 − 5i 2 4 + 5i 2 7 2 2 − 7i 2 2 + 7i 1 5 − 6i 2 5 + 6i 2
ap −1 0 −1 0 −1 0 −1 −1 0 0 −1 0 0 0 0 0 2 0 0
p o(Frobp) 3 − 8i 2 3 + 8i 2 5 − 8i 2 5 + 8i 1 4 − 9i 3 4 + 9i 1 1 − 10i 3 1 + 10i 2 3 − 10i 3 3 + 10i 3 7 − 8i 2 7 + 8i 2 11 3 4 − 11i 3 4 + 11i 2 7 − 10i 2 7 + 10i 3
96
ap 0 0 0 2 −1 2 −1 0 −1 −1 0 0 −1 −1 0 0 −1
Table 6.9: Galois group D3 , level n = 13 + 28i, considered mod 5 and mod 7
p o(Frobp) 1+i 3 1 − 2i 2 1 + 2i 3 3 2 2 − 3i 1 2 + 3i 3 1 − 4i 2 1 + 4i 2 2 − 5i 3 2 + 5i 3 1 − 6i 3 1 + 6i 3 4 − 5i 2 4 + 5i 2 7 3 2 − 7i 2 2 + 7i 2 5 − 6i 3 5 + 6i 2
ap −1 0 −1 0 2 −1 0 0 −1 −1 −1 −1 0 0 −1 0 0 −1 0
p o(Frobp) 3 − 8i 2 3 + 8i 2 5 − 8i 2 5 + 8i 3 4 − 9i 3 4 + 9i 2 1 − 10i 1 1 + 10i 2 3 − 10i 2 3 + 10i 2 7 − 8i 2 7 + 8i 2 11 2 4 − 11i 3 4 + 11i 2 7 − 10i 2 7 + 10i 1
97
ap 0 0 0 −1 −1 0 2 0 0 0 0 0 0 −1 0 0 2
Table 6.10: Galois group D3 , level n = 8 + 35i, considered mod 5 and mod 7
p o(Frobp) 1+i 2 1 − 2i 3 1 + 2i 3 3 2 2 − 3i 2 2 + 3i 3 1 − 4i 3 1 + 4i 2 2 − 5i 2 2 + 5i 2 1 − 6i 3 1 + 6i 2 4 − 5i 3 4 + 5i 1 7 3 2 − 7i 3 2 + 7i 1 5 − 6i 2 5 + 6i 3
ap 0 −1 −1 0 0 −1 −1 0 0 0 −1 0 −1 2 −1 −1 2 0 −1
p o(Frobp) 3 − 8i 3 3 + 8i 1 5 − 8i 2 5 + 8i 3 4 − 9i 3 4 + 9i 2 1 − 10i 1 1 + 10i 3 3 − 10i 2 3 + 10i 3 7 − 8i 1 7 + 8i 2 11 2 4 − 11i 2 4 + 11i 2 7 − 10i 2 7 + 10i 3
98
ap −1 2 0 −1 −1 0 2 −1 0 −1 2 0 0 0 0 0 −1
6.5.2 Dihedral Group D5 In Section 4.3.2 we computed a mod 11 representation ρ with level n = 19 + 20i and quadratic character ε19+20i , which factors through a Galois extension L over K = Q(i) with G = Gal(L/K) ∼ = D5 . In the following table we recall the correspondence between orders of elements of G and the traces of the images of those elements under ρ. order
trace
1
2
2
0
5
3, 7
There are actually two representations: wherever one has a 3 or a 7, the other will have the opposite. Both forms, denoted {ap } and {˜ ap } below, were found for the predicted weights for which we looked. In the following table we list, for some small primes p, the order of Frobp as discussed above along with the coefficients ap and a ˜p of the corresponding systems of eigenvalues found. So far we have only checked two of the three predicted weights, but found both forms in each. Table 4.7 in Section 4.3.2 lists the predicted weights along with the level, character and ` for each modular form predicted by the conjecture to correspond to ρ.
99
Table 6.11: Galois group D5 , level n = 19 + 20i, considered mod 11
p o(Frobp) 1+i 5 1 − 2i 5 1 + 2i 5 3 2 2 − 3i 2 2 + 3i 5 1 − 4i 2 1 + 4i 2 2 − 5i 2 2 + 5i 2 1 − 6i 5 1 + 6i 5 4 − 5i 2 4 + 5i 2 7 2 2 − 7i 2 2 + 7i 5 5 − 6i 5 5 + 6i 2
ap 7 7 3 0 0 3 0 0 0 0 7 7 0 0 0 0 3 7 0
a˜p 3 3 7 0 0 7 0 0 0 0 3 3 0 0 0 0 7 3 0
p o(Frobp) 3 − 8i 2 3 + 8i 5 5 − 8i 2 5 + 8i 2 4 − 9i 2 4 + 9i 5 1 − 10i 5 1 + 10i 5 3 − 10i 1 3 + 10i 2 7 − 8i 2 7 + 8i 2 4 − 11i 5 4 + 11i 5 7 − 10i 5 7 + 10i 5
100
ap 0 7 0 0 0 3 7 3 2 0 0 0 7 3 3 3
a˜p 0 3 0 0 0 7 3 7 2 0 0 0 3 7 7 7
6.6
Final Remarks
In summary, the computational evidence presented here supports the surmise that the BDJ conjectural weight recipe for totally real fields will hold in the case of imaginary quadratic fields as well. For the examples of Galois representations computed here, corresponding modular forms were found in almost all of the predicted weights. There were two exceptions. In one example we did not find the form in all the weights because the computations were too large for the program, so we have not yet looked for all of the predicted weights. In the other exception, we found the form in only two of the four predicted weights. It does look like we may have found a twist of the form in the remaining two weights. Further investigation is required. The modular symbols computation method used in my program is justified here only for K = Q(i). I expect it will be straightforward to use the same methodology for all the other Euclidean class number one imaginary quadratic fields. Note, however, that for each field one must justify an algebraic proposition such as the one presented in Section 5.2. We already know the relations we expect to work for each of these fields, namely those relations computed by Cremona, et al. See, for example, [Cre84]. Several students of Cremona (namely Bygott [Byg98], Lingham [Lin05] and Whitley [Whi90]) have extended the modular symbols method (for trivial weights) to imaginary quadratic fields of higher class number. I expect, with some work, that the method presented in this thesis can be joined with their work to compute modular forms with arbitrary weight for imaginary quadratic fields of higher class number.
101
References [ADP02] A. Ash, D. Doud, and D. Pollack, Galois representations with conjectural connections to arithmetic cohomology, Duke Mathematical Journal 112 (2002), 521–579. [AS00] A. Ash and W. Sinnott, An analogue of Serre’s conjecture for Galois representations and Hecke eigenclasses in the mod p cohomology of GL(n, Z), Duke Mathematical Journal 105 (2000), 1–24. [Ash94] A. Ash, Unstable cohomology of SL(n, O), Journal of Algebra 167 (1994), 330–342. [BCP97] W. Bosma, J. Cannon, and C. Playoust, The Magma algebra system. I. The user language, Journal of Symbolic Computation 24 (1997), no. 3-4, 235–265. [BDJ]
K. Buzzard, F. Diamond, and F. Jarvis, On Serre’s conjecture for mod ` Galois representations over totally real fields, preprint.
[BS73] A. Borel and J.-P. Serre, Corners and arithmetic groups, Commentarii Mathematici Helvetici 48 (1973), 436–491. [Byg98] J. Bygott, Modular forms over imaginary quadratic number fields, Ph.D. thesis, Exeter University, 1998. [Cre84] J. Cremona, Hyperbolic tessellations, modular symbols, and elliptic curves over complex quadratic fields, Compositio Mathematica 51 (1984), 275–323. [S¸08]
M. H. S¸eng¨ un, Serre’s conjecture over imaginary quadratic fields, Ph.D. thesis, University of Wisconsin-Madison, 2008. 102
[DDR] L. Demb´el´e, F. Diamond, and D. Roberts, Numerical examples and evidence for Serre’s conjecture over totally real fields, preprint. [DF+ 97] M. Daberkow, C. Fieker, et al., Kant v4, Journal of Symbolic Computation 24 (1997), 267–283. [Dia97] F. Diamond, The refined conjecture of Serre, Elliptic Curves and Fermat’s Last Theorem (J. Coates and S.-T. Yau, eds.), International Press, Hong Kong, 1997, 2nd. ed., pp. 172–186. [Fig95] L. M. Figueiredo, Serre’s conjecture over imaginary quadratic fields, Ph.D. thesis, University of Cambridge, 1995. [Fig99]
, Serre’s conjecture for imaginary quadratic fields, Compositio Mathematica 118 (1999), 103–122.
[Gee] [Gee07]
T. Gee, On the weights of mod p Hilbert modular forms, preprint. , Companion forms over totally real fields II, Duke Mathematical Journal 136 (2007), 275284.
[Her]
F. Herzig, The weight in a Serre-type conjecture for tame n-dimensional Galois representations, preprint.
[Jan96] G. Janusz, Algebraic number fields, 2 ed., Graduate Studies in Mathematics, vol. 7, American Mathematical Society, USA, 1996. [KM]
J. Kl¨ uners and G. Malle, A database for number fields, available from http://www.math.uni-duesseldorf.de/ klueners/minimum/minimum.html.
[KWa] C. Khare and J.-P. Wintenberger, Serre’s modularity conjecture (I), preprint. [KWb]
, Serre’s modularity conjecture (II), preprint.
[Lin05] M. Lingham, Modular forms and elliptic curves over imaginary quadratic fields, Ph.D. thesis, University of Nottingham, 2005.
103
[LS76]
R. Lee and R. H. Szczarba, On the homology and cohomology of congruence subgroups, Inventiones Mathematicae 33 (1976), no. 1, 15–53.
[Mar01] F. Martin, P´eriodes de formes modulaires de poids 1, Ph.D. thesis, Universit´e Paris 7, 2001. [Mil08] J. S. Milne, Algebraic number theory, 2008, Version 3.00, available from http://www.jmilne.org/math/CourseNotes/ANT.pdf. [Sch]
M. Schein, Weights in Serre’s conjecture for Hilbert modular forms: the ramified case, preprint.
[Ser77] J.-P. Serre, Linear representations of finite groups, Graduate Texts in Mathematics, vol. 42, Springer, New York, 1977. ¯ , Sur les repr´esentations modulaires de degr´e 2 de Gal(Q/Q), Duke
[Ser87]
Mathematical Journal 54 (1987), 179–230. [Sil86]
J. Silverman, The arithmetic of elliptic curves, Graduate Texts in Mathematics, vol. 106, Springer-Verlag, New York, 1986.
[Ste]
W.
Stein,
Sage
mathematical
software
system,
available
from
http://modular.math.washington.edu/sage/doc/reference/. [The05] The PARI Group, Bordeaux, PARI/GP, version 2.3.1, 2005, available from http://pari.math.u-bordeaux.fr/. [Whi90] E. Whitley, Modular forms and elliptic curves over imaginary quadratic fields, Ph.D. thesis, Exeter University, 1990. [Wie05] G. Wiese, Modular forms of weight one over finite fields, Ph.D. thesis, Universiteit Leiden, 2005.
104
